Semi-parametric approach to large-scale portfolio optimization with factor models of asset returns

ABSTRACT

An approach to large-scale portfolio optimization for asset returns represented by factor models is disclosed. Factor models can be used within general portfolio optimization problems, such as mean-variance optimization, expected utility maximization, and mean-risk optimization, with various measures of risk, including conditional Value-at-Risk, as well as the representation of risk constraints and constraints on higher moments of the asset return distribution. Both expected utility maximization and mean-risk optimization are more general than mean-variance optimization and can consider fat tails in the asset return distribution and, thus, allow for better control of downside risk. Explicit risk constraints especially constraints on conditional Value-at-Risk, limit downside risk in either mean-variance optimization, expected utility maximization, or mean-risk optimization. Constraints on higher moments limit fat tails of the asset return distribution. Equilibrium returns in expected utility maximization and mean-variance optimization based on factor models of asset returns are obtained. Active management of portfolios of financial assets based on factor exposures is provided.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates generally to a system and method formanagement of a portfolio of financial assets and, more particularly, tolarge-scale portfolio optimization. Specifically, various embodiments inaccordance with the present invention provide a system and method formodeling and solving large-scale portfolio optimization problems,including mean-variance optimization, expected utility maximization, andgeneral mean-risk optimization problems.

Description of the Prior Art

Since H. Markowitz, Portfolio selection, Journal of Finance, 7(1):77-91, 1952, portfolio management problems are routinely formulated andsolved as mean-variance portfolio optimization problems, where theexpected return of a portfolio is traded off with its risk and whererisk is represented as portfolio variance. Let {tilde over (R)} be therandom n-vector of asset returns. The mean-variance portfoliooptimization problem may be stated as

${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}x^{T}M\; x}$Ax = b, l ≤ x ≤ h

where E{tilde over (R)} is the n-vector of expected asset returns;M=[M_(ij)] is the n×n covariance matrix of asset returns(M_(ij)=cov({tilde over (R)}_(i),{tilde over (R)}_(j))), γ is the riskaversion parameter, Ax=b are linear constraints, and l and h are lowerand upper bounds on asset holdings. The formulation Ax=b may include aportfolio constraint e^(T)x=1, for modeling both long-only andlong-short portfolios and/or more elaborate constraints for controllingthe leverage of long-short portfolios. It may also include sectorexposure constraints, industry exposure constraints, transaction costmodeling, turnover constraints, and any constraints related to apiecewise-linear market impact model representation.

Using mean-variance optimization is particularly appropriate when assetreturns are approximately multi-variate normally distributed, i.e.,{tilde over (R)}≈N(E{tilde over (R)}, M), since in this case thedistribution is fully determined by E{tilde over (R)} and M only and allhigher moments are either zero (odd) or monotonically determined by M(even). But it is also appropriate to use when asset returns are notapproximately multi-variate normally distributed but an investor onlycares about portfolio variance (or tracking error) as a measure of risk.By varying the risk aversion parameter γ from zero to a very largenumber, one may determine the optimal efficient frontier from the pointof considering expected returns only (γ=0) to the point of the minimumvariance portfolio (γ=∝).

A related, more general, concept is expected utility maximization. Letu(W) be a concave utility function of wealth (W). The expected utilitymaximization problem may be stated as

max E u(1+{tilde over (R)} ^(T) x)

Ax=b,l≤x≤h

where, given an initial wealth normalized to 1, the end-of-period wealthis given by the random variable W=1+{tilde over (R)}^(T)x and evaluatedby the utility function u(W). Typically, the utility function is assumedto be monotonically increasing and concave. The functional form of theutility function u(W), in particular, the ArrowPratt risk aversion; seeK. J. Arrow, Aspects of the theory of risk bearing, Essays on the Theoryof Risk Bearing, Markham, Chicago, pages 90-109, 1965 and J. W. Pratt,Risk aversion in the small and in the large, Econometrica, 32, pages122-136, 1964, a measure of the second derivative of the utilityfunction, represents the investor preference. Utility functionsfrequently used are from the HARA (hyperbolic absolute risk aversion)class of utility functions. Often in finance the power function is used,i.e.,

${{u(W)} = \frac{W^{1 - \gamma} - 1}{1 - \gamma}},$

where γ≥0 and γ≠1. Here γrepresents the constant (with respect to wealth) relative risk aversionparameter describing the investor preference towards risk. Therefore,the power function is typically referred to as CRRA (constant relativerisk aversion). Another utility function in the HARA class is theexponential utility function u(W)=−e^(−λW), where λ represents theconstant (with respect to wealth) absolute risk aversion parameter. Theexponential utility function is also referred to as CARA (constantabsolute risk aversion). The logarithmic utility function u(W)=log(W) isa special case of the power function for the limit of the risk, aversionγ=1. The latter has the property of maximizing growth in a multi-periodsetting. A generalization is the generalized log utility functionu(W)=log(a+W), where, by proper choice of a, increasing and decreasingrelative risk aversion may be modeled; see M. Rubin-stein, Risk aversionin the small and in the large, Econometrics, pages 32:122-136, 1965. Theabove-mentioned functions exhaust the class of HARA utility functions.An one-switch class of utility functions is given by D. Bell, One switchutility functions and a measure of risk. Management Science,24(12):1416-1424, 1965. Other utility functions that are not necessarilymonotonically increasing and concave have been devised to representspecific investor risk preferences; see D. Kahnemann and A. Tversky,Prospect theory: An analysis of decisions under risk, Econometrics,47(2), pages 263-291, 1979. These are important for determiningindividualized portfolios.

Expected utility maximization facilitates the appropriate representationof all higher moments (skewness, kurtosis, etc.) of the asset returndistribution in the portfolio optimization framework. It can be shownthat if the utility function is quadratic (using only the increasingpart of the quadratic function), the expected utility maximizationproblem results in a mean-variance optimization problem and thereforegives identical results. Any other utility function will yield differentresults, when asset returns are not multi-variate normally distributed.If asset returns are multi-variate normally distributed, anymonotonically increasing and concave utility function will yield amean-variance efficient portfolio. One expects different results forexpected utility maximization than mean-variance analysis when assetreturns are not multi-variate normally distributed and the utilityfunction is not quadratic. But the differences in the portfolios may besmall, as Y. Kroll, H. Levy, and H. Markowitz, Mean-variance versusdirect utility maximization, Journal of Finance, 39(1), pages 47-61,1984, argued. Since one may locally approximate any utility function bya quadratic approximation, the mean-variance model will in most casesgive reasonable results even for asset returns not following amulti-variate normal distribution. This explains the great success ofmean-variance portfolio optimization. However, tail behavior might bevery different.

A generalization of the mean-variance framework is mean-risk portfoliooptimization. The mean-risk portfolio optimization problem may be statedas

${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}\mspace{11mu} {{Risk}( {{\overset{\sim}{R}}^{T}x} )}}$Ax = b, l ≤ x ≤ h

where Risk({tilde over (R)}^(T)x) is a risk measure on the distributionof portfolio returns {tilde over (R)}^(T)x. In mean-variance portfoliooptimization the risk measure used is portfolio variance, i.e.,Risk({tilde over (R)}^(T)x)=var({tilde over (R)}^(T)x)=E({tilde over(R)}^(T)x−E({tilde over (R)}^(T)x))²=x^(T)Mx. Mean-variance optimizationis therefore part of the broader class of mean-risk portfoliooptimization, where various risk measures are considered.

Risk measures are typically either dispersion measures or downside riskmeasures. Variance is a dispersion measure. Another dispersion measureis the mean-absolute-deviation measure (MAD), introduced by H. Konno andH. Yamazaki, Mean-absolute deviation portfolio optimization model andits application to Tokyo stock market, Management Science, 37(5), pages519-531, 1991. Typical downside risk measures are the semi-variance andvariants of lower partial moments. But most important in finance are thetail measures Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR).

In addition to considering a risk measure in the objective function,risk, especially downside risk, may also be controlled as a constraint.For example,

Risk({tilde over (R)} ^(T) x)≤ρ

may be a constraint added to the mean-variance portfolio optimizationproblem or to the expected utility portfolio optimization problem, whereρ is the maximum level of acceptable risk as defined by the risk measureRisk. Special risk constraints may involve higher moments of the returnsdistribution, in particular, one may constrain the skewness and/or thekurtosis of the returns distribution.

SUMMARY OF THE INVENTION

Various embodiments in accordance with the present invention provide asystem and method for efficiently solving general classes of large-scalefinancial asset portfolio optimization problems. Preferred embodimentsof the system and method in accordance with the present invention solvethe general portfolio optimization problem, employing a factorrepresentation of asset returns. Various embodiments in accordance withthe present invention calibrate the optimization model to a benchmark toobtain unconditional mean returns and enable active management based onconditional expected return predictions. Various additional embodimentsof the system and method in accordance with the present inventionconsider derivatives as part of the portfolio.

BRIEF DESCRIPTION OF THE DRAWING

The various embodiments of the present invention will be described inconjunction with the accompanying figures of the drawing to facilitatean understanding of the present invention. In the figures, likereference numerals refer to like elements. In the drawing:

FIG. 1 is a block diagram of an example of a system in accordance with apreferred embodiment of the present invention implemented on a personalcomputer.

FIG. 2 is a block diagram of an example of a system in accordance withan alternative embodiment of the present invention implemented on apersonal computer coupled to a web or Internet server.

FIG. 3 is a flowchart illustrating a method in accordance with apreferred example of the present invention for providing large-scaleportfolio optimization with factor models of asset returns.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention is particularly applicable to a computerimplemented software based financial asset portfolio management systemfor providing large-scale portfolio optimization, and it is in thiscontext that the various embodiments of the present invention will bedescribed. It will be appreciated, however, that the system and methodfor providing general portfolio optimization, including mean-varianceoptimization, expected utility maximization, and mean-risk optimizationin large-scale portfolio management in accordance with the presentinvention have greater utility, since they may be implemented inhardware or may incorporate other modules or functionality not describedherein.

FIG. 1 is a block diagram illustrating an example of a generalportfolio, management system 10 for large-scale portfolio optimizationin accordance with one embodiment of the present invention implementedon a personal computer 12. In particular, the personal computer 12 mayinclude a display unit 14, which may be a cathode ray tube (CRT), aliquid crystal display, or the like; a processing unit 16; and one ormore input/output devices 18 that permit a user to interact with thesoftware application being executed by the personal computer. In theillustrated example, the input/output devices 18 may include a keyboard20 and a mouse 22, but may also include other peripheral devices, suchas printers, scanners, and the like. The processing unit 16 may furtherinclude a central processing unit (CPU) 24, a persistent storage device26, such as a hard disk, a tape drive, an optical disk system, aremovable disk system, or the like, and a memory 28. The CPU 24 maycontrol the persistent storage device 26 and memory M. Typically, asoftware application may be permanently stored in the persistent storagedevice 26 and then may be loaded into the memory 28 when the softwareapplication is to be executed by the CPU 24. In the example shown, thememory 28 may contain a large-scale portfolio optimization tool 30 forportfolio management. The portfolio optimization tool 30 may beimplemented as one or more software modules that are executed by the CPU24. In accordance with various contemplated embodiments of the presentinvention, the general portfolio management system 10 may also beimplemented using hardware and may be implemented on different types ofcomputer systems, such as client/server systems, Web servers, mainframecomputers, workstations, and the like.

Thus, in accordance with another embodiment of the present invention,the general portfolio optimization system 10 is implemented via a hostedWeb server. A system using a hosted Web server, generally indicated bythe numeral 1801, is shown in FIG. 2. The system 1801 preferablycomprises a Web-based application accessed by a personal computer 1802,as shown in FIG. 2. For example, the personal computer 1802 may be anypersonal computer having at least two gigabytes of random access memory(RAM), using a Web browser, preferably MICROSOFT Internet Explorer 6.0browser or greater. In this example, the system 1801 is a 128-bit SSLencrypted secure application running on a MICROSOFT Windows Server 2003operating system or Windows Server 2000 operating system or lateroperating system available from Microsoft Corporation located inRedmond, Wash. The personal computer 1802 also, comprises a hard diskdrive preferably having at least 40 gigabytes of free storage spaceavailable. The personal computer 1802 is coupled to a network 1807. Forexample, the network 1807 may be implemented using an Internetconnection. In one implementation of the system 1801, the personalcomputer 1802 can be ported to the Internet or Web, and hosted by aserver 1803. The network 1807 may be implemented using a broadband dataconnection, such as, for example, a DSL or greater connection, and ispreferably a Tl or faster connection. The graphical user interface ofthe system 1801 is preferably displayed on a monitor 1804 connected tothe personal computer 1802. The monitor 1804 comprises a screen 1805 fordisplaying the graphical user interface provided by the system 1801. Themonitor 1804 may be a 15 color monitor and is preferably a 1024×768,24-bit (16 million colors) VGA monitor or better. The personal computer1802 further comprises a 256 or more color graphics video card installedin the personal computer. As shown in FIG. 2, a mouse 1806 is providedfor mouse-driven navigation between screens or windows comprising thegraphical user interface of the system 1801. The personal computer 1802is also preferably connected to a keyboard 1808. The mouse 1806 andkeyboard 1808 enable a user utilizing the system 1801 to perform generalportfolio management. Preferably, the user can print the results using aprinter 1809. The system 1801 is implemented as a Web-based application,and data may be shared with additional software (e.g., a word processor,spreadsheet, or any other application). Persons skilled in the art willappreciate that the systems and techniques described herein areapplicable to a wide array of business and personal applications.

In accordance with a preferred example of the method of the presentinvention shown in FIG. 3 to manage a portfolio of financial assets toprovide large-scale portfolio optimization, including mean-varianceoptimization, expected utility maximization, and general mean-riskportfolio optimization, the representation of asset returns is presentedvia a factor model.

Asset Returns

Factor models have been introduced that linearly relate at each periodt≥1 the n-vector of asset returns {tilde over (R)}_(t) to the values (orchange in values) of a smaller number k of factors, {tilde over(V)}_(t). The factor model representation of the asset returns has manydesirable properties, including that it has good explanatory power andthe resulting covariance matrix of asset returns is of full rank. It hasalso theoretical importance, as modern asset pricing theories havefactor models as their underpinnings, e.g., the capital asset pricingmodel (CAPM); see William F. Sharpe, Capital asset prices: A theory ofmarket equilibrium under conditions of risk, Journal of Finance, 19(3):pages 425-442, 1964 and John Liirtner, The valuation of risk assets andthe selection of risky investments in stock portfolios and capitalbudgets, Review of Economics and Statistics, 47(1): pages 13-37, 1965;and the arbitrage pricing theory (APT); see Stephen A. Ross, Thearbitrage theory of capital asset pricing; Journal of Economic Theory,13 (3), pages 341-360, provide equilibrium prices and returns for assetstraded in the markets.

Suppose asset returns in each period t≥1 follow a factor model,

{tilde over (R)} _(t) ={tilde over (F)} _(t) ^(T) {tilde over (V)}_(t)+{tilde over (ε)}_(t)

where {tilde over (F)}_(t) is the k×n random matrix of factor loadings,{tilde over (V)}_(t) is the random k-vector of the values of the factors(sometimes also called factor returns), and {tilde over (ε)}_(t) is therandom n-vector of idiosyncratic returns. The formulation includes amean (or intercept) vector, if we define the (random) value of the firstfactor as having always the value 1, thus, the random returns for eachasset i are represented as

{tilde over (R)} _(it) ={tilde over (F)} _(1it) +{tilde over (F)} _(2it){tilde over (V)} _(2t) + . . . +{tilde over (F)} _(kit) {tilde over (V)}_(kt)+{tilde over (ε)}_(it).

We assume that the idiosyncratic returns {tilde over (ε)}_(t) aremulti-variate normally distributed, {tilde over (ε)}=N(0, Σ_(t)), wherethe covariance Σ_(t)=diag(σ_(it) ²), and {tilde over (ε)}_(t) is assumedindependently distributed, between its components, respectively, andindependently distributed with respect to {tilde over (V)}_(t). (Afactor model may also be defined for asset risk premia, i.e., excessreturns over the risk-free rate, and the risk-free rate may be added tothe asset risk premia to obtain asset returns.)

Underlying the factor model of asset returns may be two differentstatistical models of asset returns:

Statistical model (1): Let {tilde over (F)}_(t)=F be constant. Let{tilde over (V)}_(t), t≥1 and {tilde over (ε)}_(t), t≥1 be eachindependently and identically distributed random variables. Then {tildeover (R)}_(t)=F^(T) {tilde over (V)}_(t)+{tilde over (ε)}_(t), t≥1 is anindependently and identically distributed random variable. We observe ateach period t=1, . . . T an outcome R_(t), V_(t), and ε_(t) of {tildeover (R)}_(t), {tilde over (V)}_(t), and {tilde over (ε)}_(t),respectively. At period T+1, the current period at which a portfoliodecision is to be made, we write the random vector of asset returns as

{tilde over (R)} _(T+1) |{tilde over (R)}1, . . . ,{tilde over (R)}T=F^(T) {tilde over (V)} _(T+1) |{tilde over (V)}1, . . . ,{tilde over(V)}T+{tilde over (ε)}_(T+1)|{tilde over (ε)}1, . . . ,{tilde over(ε)}T.

But based on independence, {tilde over (R)}_(T+1)={tilde over(R)}_(T+1)|{tilde over (R)}₁, . . . , {tilde over (R)}_(T), {tilde over(V)}_(T+1)={tilde over (V)}_(T+1)|{tilde over (V)}₁, . . . , {tilde over(V)}_(T) and {tilde over (ε)}_(T+1)={tilde over (ε)}_(T−1)|{tilde over(ε)}₁, . . . , {tilde over (ε)}_(T), thus,

{tilde over (R)} _(T+1) =F ^(T) {tilde over (V)} _(T+1)+{tilde over(ε)}_(T+1),

which we may write as

{tilde over (R)}=F ^(T) {tilde over (V)}+{tilde over (ε)}

by setting {tilde over (R)}≡{tilde over (R)}_(T−1), {tilde over(V)}≡{tilde over (V)}_(T+1), and {tilde over (ε)}≡{tilde over(ε)}_(T+1), thereby suppressing the time index for period T+1.Accordingly, {tilde over (ε)}=N(0, Σ), where Σ=diag(σ_(i) ²).

Statistical model (1) is applicable to macro-economic factor models. Thefactors in this framework may be macro-economic variables that influenceasset returns, such as (changes in) gross domestic product, oil prices,unemployment rate, interest rates, etc., and the factor loadings of anasset represent its exposure to each of the macro-economic factors.

Extensions of the model include possible time dependency of {tilde over(V)}_(t) and/or {tilde over (ε)}_(t), by defining the conditionaldistributions {tilde over (R)}_(T+1)|{tilde over (R)}₁, . . . , {tildeover (R)}_(T), {tilde over (V)}_(T+1)|{tilde over (V)}₁, . . . , {tildeover (V)}_(T) and/or {tilde over (ε)}_(T+1)|{tilde over (ε)}₁, . . . ,{tilde over (ε)}_(T). For example, considering {tilde over(V)}_(T+1)|{tilde over (V)}₁, . . . , {tilde over (V)}_(T) includestime-series models of the factors such as vector autoregressiveprocesses and considering {tilde over (ε)}_(T+1)|{tilde over (ε)}₁, . .. , {tilde over (ε)}_(T) includes models with time-varying idiosyncraticvariances, i.e., GARCH processes.

Statistical model (2): Let {tilde over (F)}_(t), t≥1 be a sequence ofindependently and identically distributed random variables. Conditionalon {tilde over (F)}_(t), let {tilde over (V)}_(t) and {tilde over(ε)}_(t), t≥1 be each independently and identically distributed randomvariables. Then, {tilde over (R)}_(t)|{tilde over (F)}_(t)={tilde over(F)}_(t) ^(T){tilde over (V)}_(t)|{tilde over (F)}_(t)+{tilde over(ε)}_(t)|{tilde over (F)}_(t) is an independently and identicallydistributed random variable. We observe at each period t=1, . . . T anoutcome R_(t), V_(t), F_(t), and ε_(t) of {tilde over (R)}_(t), {tildeover (V)}_(t), {tilde over (F)}_(t), and {tilde over (ε)}_(t),respectively and at the current period T+1. at which a portfoliodecision is to be made, an outcome F_(t) of {tilde over (F)}_(t). At thecurrent period T÷1 we write the random vector of asset returns as

{tilde over (R)} _(T+1) |{tilde over (R)} ₁ ,{tilde over (F)} ₁ , . . .,{tilde over (R)} _(T) ,{tilde over (F)} _(T) ,{tilde over (F)} _(T+1)={tilde over (F)} _(T÷1) {tilde over (V)} _(T+1) |{tilde over (V)}₁,{tilde over (ε)}₁ , . . . ,{tilde over (V)} _(T) ,{tilde over (F)}_(T) ,{tilde over (F)} _(T+1)+{tilde over (ε)}_(T+1)|{tilde over (ε)}₁,{tilde over (F)} ₁, . . . ,{tilde over (ε)}_(T) ,{tilde over (F)} _(T),{tilde over (F)} _(T+1).

Based on independence,

{tilde over (R)} _(T÷1) |{tilde over (F)} _(T+1) ={tilde over (R)}_(T+1) |{tilde over (R)} ₁ ,{tilde over (F)} ₁ , . . . ,{tilde over (R)}_(T) {tilde over (F)} _(T) ,{tilde over (F)} _(T+1),

V _(T+1) |{tilde over (F)} _(T+1) ={tilde over (V)} _(T+1) |{tilde over(V)} ₁ ,{tilde over (F)} ₁ , . . . ,{tilde over (V)} _(T) ,{tilde over(F)} _(T) ,{tilde over (F)} _(T+1)

and

{tilde over (ε)}_(T÷1) |{tilde over (F)} _(T+1)={tilde over(ε)}_(T+1)|{tilde over (ε)}₁ ,{tilde over (F)} ₁, . . . ,{tilde over(ε)}_(T) ,{tilde over (F)} _(T) ,{tilde over (F)} _(T+1),

thus,

{tilde over (R)} _(T+1) |{tilde over (F)} _(T+1) ={tilde over (F)}_(T+1) ^(T) {tilde over (V)} _(T+1) |{tilde over (F)} _(T+1)+{tilde over(ε)}_(T+1) |{tilde over (F)} _(T+1).

Since at period T+1, an outcome F_(T+1) of {tilde over (F)}_(T+1) isobserved, we may write

{tilde over (R)}=F ^(T) {tilde over (V)}+{tilde over (ε)}

by setting {tilde over (R)}≡{tilde over (R)}_(T+1)|{tilde over(F)}_(T+1)=F_(T+1), {tilde over (V)}≡{tilde over (V)}_(T+1)|{tilde over(F)}_(T+1)=F_(T+1), {tilde over (ε)}≡{tilde over (ε)}_(T+1)|{tilde over(F)}_(T+1)=F_(T+1), and F≡F_(T÷1), thereby suppressing the time indexT+1 and the dependency on the observed value F_(T+1) of {tilde over(F)}_(T÷1). Accordingly, {tilde over (ε)}=N(0, Σ), where Σ=diag(σ_(i)²).

Statistical model (2) is applicable to fundamental factor models, wherefactor loadings may be asset-specific fundamental quantities, say, dataderived from accounting statements, such as earnings (over price ratio),dividend yield, past performance, etc., and a factor may be defined asthe return of a (long-short) portfolio that has an exposure of one to aspecific factor loading and a zero exposure to all other factor loadingsconsidered in the factor model. Extensions of the model include possibletime dependency of {tilde over (V)}_(T+1) and/or {tilde over (ε)}_(T÷1),by, defining the conditional distributions {tilde over (R)}≡{tilde over(R)}_(T+1)|{tilde over (R)}₁, {tilde over (F)}₁, . . . , {tilde over(R)}_(T), {tilde over (F)}_(T), {tilde over (F)}_(T+1), {tilde over(V)}≡{tilde over (V)}_(T+1)|{tilde over (V)}₁, {tilde over (F)}₁, . . ., {tilde over (V)}_(T), {tilde over (F)}_(T), {tilde over (F)}_(T+1)and/or {tilde over (ε)}≡{tilde over (ε)}_(T+1)|{tilde over (ε)}₁, {tildeover (F)}₁, . . . , {tilde over (ε)}_(T), {tilde over (F)}_(T), {tildeover (F)}_(T+1). For example, defining {tilde over (V)}_(T+1)|{tildeover (V)}₁, {tilde over (F)}₁, . . . , {tilde over (V)}_(T), {tilde over(F)}_(T), {tilde over (F)}_(T+1) would allow for time-series models ofthe factors and/or defining {tilde over (ε)}_(T+1)|{tilde over (ε)}₁,{tilde over (F)}₁, . . . , {tilde over (ε)}_(T), {tilde over (F)}_(T),{tilde over (F)}_(T+1) would allow for models with time-varyingidiosyncratic variances, i.e., GARCH processes.

The factor model representation {tilde over (R)}=F^(T){tilde over(V)}+{tilde over (ε)}, based on statistical models (1) and (2), covers awide range of factor models that have been developed for representingasset returns in practical situations.

In any portfolio optimization problem, one needs to maximize theexpected value of a function of the portfolio return, say, max EG({tildeover (R)}^(T)x). In mean-variance portfolio optimization this wouldresult in max

${{E\; {\overset{\sim}{R}}^{T}x} - {\frac{\gamma}{2}{{var}( {{\overset{\sim}{R}}^{T}x} )}}},$

in utility maximization max Eu(1+{tilde over (R)}^(T)x), and inmean-risk optimization max

${E\; {\overset{\sim}{R}}^{T}x} - {\frac{\gamma}{2}{{{Risk}( {{\overset{\sim}{R}}^{T}x} )}.}}$

For asset returns following a factor model,

EG({tilde over (R)} ^(T) x)=EG((F ^(T) {tilde over (V)}+{tilde over(ε)})^(T) x)

and calculating this expectation, without taking advantage of anyspecial structure, involves multiple integration:

${{{EG}( {{\overset{\sim}{R}}^{T}x} )} = {\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}\mspace{11mu} {\ldots  {\int_{- \infty}^{+ \infty}{{G( {{( {F^{T}\overset{\sim}{V}} )^{T}x} + {\sum\limits_{i}\; {\sigma_{i}x_{i}z_{i}}}} )}{{dP}( \overset{\sim}{V} )}{p_{1}( z_{1} )}\mspace{14mu} \ldots \mspace{11mu} {p_{n}( z_{n} )}{dz}_{1\mspace{11mu}}\ldots \mspace{11mu} {dz}_{n}}}}}}},$

where P({tilde over (V)}) is the cumulative distribution function of thefactors {tilde over (V)} and where z_(i)=N(0,1) is an independent unitnormal distribution with density function

${{p_{i}( z_{i} )} = {\frac{1}{\sqrt{2\pi}}e^{{- z_{i}^{2}}/2}}},$

for each asset i=1, . . . , n. In this formulation, portfolio returnsare a linear function of the portfolio weights x.

In order to maximize EG({tilde over (R)}^(T)x) one needs functionevaluations and gradients as a function of the portfolio weights x or atany given value of x.

For a given portfolio with weights x, the factor model returns of aportfolio,

{tilde over (R)} _(T) x=(F ^(T) x{tilde over (V)}+{tilde over (ε)})^(T)x=(F ^(T) {tilde over (V)})^(T) x+{tilde over (ε)} ^(T) x,

may be expressed as

{tilde over (R)} ^(T) x=(F ^(T) {tilde over (V)})^(T) x+σ(x)z,

based on the assumption that the idiosyncratic returns are eachindependently normally distributed, from which

${\sigma (x)} = {\sqrt{x^{T}\Sigma \; x} = \sqrt{\sum\limits_{i = 1}^{n}\; {\sigma_{i}^{2}x_{i}^{2}}}}$and z = N(0, 1)

is a one dimensional unit normal distribution. Thus,

EG({tilde over (R)} ^(T) x)=∫_(−∞) ^(+∞)∫_(−∞) ^(+∞) G((F ^(T) {tildeover (V)})^(T) x+σ(x)z)dP({tilde over (V)})p(z)dz,

where

${p(z)} = {\frac{1}{\sqrt{2\pi}}e^{{- z^{2}}/2}}$

is the density of the unit normal distrubution.

With this reformulation, portfolio returns (F^(T){tilde over(V)})^(T)x+σ(x)z are a nonlinear function of the portfolio variables x,since they depend on σ(x). The function √{square root over (x^(T)Σx)} isconvex with respect to the portfolio variables x, since compoundfunctions of the type

$( {\sum_{i}{g_{i}(x)}^{q}} )^{\frac{1}{q}}$

are convex for convex and nonnegative functions g_(i)(x); see, e.g., S.P. Boyd and L. Vandenberghe, Convex Optimization, Cambridge UniversityPress, Cambridge, UK, 2004. In this context, g_(i)(x)=√{square root over(σ_(i) ²x_(i) ²)} is convex (linear) in x and nonnegative.

The expectation

EG({tilde over (R)} ^(T) x)=∫_(−∞) ^(+∞)∫_(−∞) ^(+∞) G((F ^(T) {tildeover (V)})^(T) x+σ(x)z)dP({tilde over (V)})p(z)dz,

is exactly the same as the linear form in x introduced above:

${{{EG}( {{\overset{\sim}{R}}^{T}x} )} = {\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}\mspace{11mu} {\ldots  {\int_{- \infty}^{+ \infty}{{G( {{( {F^{T}\overset{\sim}{V}} )^{T}x} + {\sum\limits_{i}\; {\sigma_{i}x_{i}z_{i}}}} )}{{dP}( \overset{\sim}{V} )}{p_{1}( z_{1} )}\mspace{14mu} \ldots \mspace{11mu} {p_{n}( z_{n} )}{dz}_{1\mspace{11mu}}\ldots \mspace{11mu} {dz}_{n}}}}}}},$

for any value of x. If two functions with respect to x have the samefunction value for any value of x, then they are the same function,irrespective of their inner workings. Thus, if one function is concave(convex) in x so is the other. From the latter expression for EG({tildeover (R)}^(T)x) it follows that EG({tilde over (R)}^(T)x) is concave(convex) if the function GO is a concave (convex) function in itsargument. This is the case, because compound functions G(h(x)) areconcave (convex) if G(·) is concave (convex) and h(x) is linear in x,and because the expectation of a concave (convex) function in x withrespect to a random variable is a concave (convex) function in x; see,e.g., S. P. Boyd and L. Vandenberghe, Convex Optimization, CambridgeUniversity Press, Cambridge, UK, 2004. Thus, we proved that

EG({tilde over (R)} ^(T) x)=∫_(−∞) ^(+∞)∫_(−∞) ^(+∞) G((F ^(T) {tildeover (V)})^(T) x+σ(x)z)dP({tilde over (V)})p(z)dz,

is a concave (convex) function with respect to x, if G(·) is a concave(convex) function in its argument.

This is not obvious, since, as stated above, (F^(T){tilde over(V)})^(T)x+σ(x)z is not concave (convex) with respect to x, becauseσ(x)z is convex for outcomes z>0 and concave for outcomes z<0. Thus,G((F^(T){tilde over (V)})^(T)x+σ(x)z) is not necessarily concave(convex) with respect to x, even if G(·) is concave (convex) in itsargument; but it may be, depending on how strongly concave (convex) G(·)is. For the convex risk measures discussed herein we have empiricallyobserved that G((F^(T){tilde over (V)})^(T)x+σ(x)z) (defined as thenegative of Risk({tilde over (R)}^(T)x)) is typically concave in x, butfor expected utility maximization with a very low risk aversionparameter, where G(·) is almost linear in its argument, G((F^(T){tildeover (V)})^(T)x+σ(x)z) may not be concave in x. This demonstrates thatthe expectation of a non concave (non convex) function can be a concave(convex) function.

We now approximate the unit normal random variable z by a discreterandom variable

ζ=(z _(v) ,p _(v))

with realizations z_(v) occurring with probability p_(v), for v=1, . . ., m. That is, the continuous unit normal distribution is represented bya histogram with the properties that its mean is zero, E(ζ)=0, itsvariance is approximately one, E(ζ²)≈1, and its higher moments matchthose of the unit normal distribution. For a sufficiently large numberof discrete outcomes, the discrete representation closely approximatesthe unit normal distribution, i.e.,

${{\lim\limits_{marrow\infty}\; {\max\limits_{z}{{{(z)} - {Q(z)}}}}} = 0},$

and the cumulative distribution function of the discrete approximation

(z) would match closely the cumulative distribution function Q(z) of theunit normal distribution.

A discrete approximation of the unit normal distribution, obtained usingoptimization, is given in Table 1. It is preferably based on 51 equallyspaced points between −5 and +5 and very closely matches the unit normaldistribution. For practical Purposes this seems sufficiently accurate,since the tail area of

${\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{- 5}{{ze}^{- \frac{z^{2}}{2}}{dz}}}} = {{2.8665e} - 07}$

of probability mass is all that is not captured on either side of theunit normal distribution. (Using 6 standard deviations, the one sidederror would be 9.8659e-10.) Its first 8 moments are: mean=0.000000,variance=1.000000, skewness=0.000000, kurtosis=3.000000, m₅=0.000000,m₆=15.000000, m₇=0.000000, m₈=105.000000.

Using the discrete approximation ζ of the unit normal random variable,we may express the returns of a portfolio generated by the factor modelfor any outcome z_(v) of ζ as

{tilde over (R)} _(v)(x)=(F ^(T) {tilde over (V)})^(T) x+σ(x)z _(v),

and compute expectations as

${{EG}( {{\overset{\sim}{R}}_{v}(x)} )} = {\int_{- \infty}^{+ \infty}{\sum\limits_{v}{{G( {{( {F^{T}\overset{\sim}{V}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{v}{{{dP}( \overset{\sim}{V} )}.}}}}$

The multi-variate distribution {tilde over (V)}, may have differentforms depending on the factors used. It may exhibit fat tails and may bepeaky. The statistics of the population of {tilde over (V)} aretypically not known and thus a parametric representation appearsdifficult. Instead, we proceed non-parametrically. Since {tilde over(V)}_(t), t=1, . . . , T+1, are assumed independently and identicallydistributed, observed outcomes V_(t) at periods t=1, . . . , T are alsoobserved outcomes of {tilde over (V)}≡{tilde over (V)}_(T+1). Let {tildeover (V)} be a random variable defined by the empirically observedoutcomes V_(t) with corresponding, probability

$p_{t} = {\frac{1}{T}.}$

Using {tilde over (V)}, we obtain an entirely discrete representation ofthe factor model returns as

R _(tv)(x)=(F ^(T) V _(t))^(T) x+σ(x)z _(v)

with associated probabilities p_(tv)=p_(t)p_(v). Defining a new discreterandom vector

(x)=(R_(tv)(x),p_(tv)), with outcomes R_(tv)(x) and associatedprobability p_(tv), and using the sample-average approximation, using{tilde over (V)}, we calculate the conditional expectation, givenζ=z_(v), as

${ {{EG}( {{\overset{\sim}{R}}_{v}(x)} )} \middle| \zeta  = {\sum\limits_{t}{{G( {{( {F^{T}V_{t}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}}}},$

where

$p_{t} = {\frac{1}{T}.}$

For sufficiently large number T of observations V_(t), EG

(x))|ζ closely approximates EG({tilde over (R)}_(v)(x))|ζ, as EG(

(x))|ζ→EG({tilde over (R)}_(v)(x))|ζ as T→∞.

Now we can calculate the expectation EG(

(x)) entirely as a multiple sum:

${{{EG}( {\overset{\sim}{R}(x)} )} = {\sum\limits_{t}{\sum\limits_{v}{{G( {{( {F^{T}V_{t}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}p_{v}}}}},$

and EG(

(x)) closely approximates EG({tilde over (R)}(x)), as EG(

(x))→EG({tilde over (R)}(x)) as T→∞ and as ζ closely approximates z.

We, thus, have, for a general factor model representation of assetreturns expressed portfolio returns as a function of x as a randomvariable with a discrete distribution. We call this approachsemi-parametric approximation, since the idiosyncratic component of theasset returns is represented parametrically and the factor explainedcomponent is represented non-parametrically. Any expectation offunctions of portfolio returns that may occur in a portfoliooptimization model can therefore, be computed by multiple sums (over tand v), making portfolio optimization problems very tractable andamenable to solution. We have shown above that EG({tilde over (R)}^(T)x)is concave (convex) in x if G(·) is concave (convex) in its argument,therefore portfolio optimization problems based on our reformulation areconvex problems that can be solved with (convex) nonlinear programmingtechniques and any local optimum is also a global optimum.

In practice, the true parameters of the distributions {tilde over (R)},{tilde over (V)}, and {tilde over (ε)} are unknown and need to beestimated. We obtain estimates of these unknown true quantities byactually estimating a factor model, based on the statistical modelsintroduced above. There are three types of factor models generally inuse: the macro-economic (see, e.g., Roll A. Chen, N. F. and S. A. Ross:Economic forces and the stock market, The Journal of Business, 59 (3):383-404, 1986, the statistical and the fundamental factor model, see,e.g., B. A. Rosenberg: Extra-market components of covariance in securityreturns, Journal of Financial and Quantitative Analysis, 9 (2): 263-273,1974. Macro-economic and statistical factor models are estimated basedon statistical model (1) and fundamental factor models are estimatedbased on statistical model (2). See G. Connor: The three types of factormodels: A comparison of their explanatory power, Financial AnalystsJournal, 51 (3): 42-46, 1995, about the explanatory power of the threetypes of factor models. There are also hybrid models as a suitablecombination of the three factor models. A related approach, includes E.F. Fama and K. R. French: The cross-section of expected stock returns,Journal of Finance, 47 (2): 427-465, 1992, and E. F. Fama and K. R.French: Common risk factors in the returns on stocks and bonds, Journalof Financial Economics, 33: 3-56, 1993, where factors are defined asreturns of so called factor-mimicking portfolios. For the estimation offactor models, see also E. F. Fama and J. D. MacBeth: Risk, return, andequilibrium: Empirical tests, The Journal of Political Economy, 81 (3):607-636, 1973.

In the following the discrete representation of the factor returns isapplied to expected utility optimization, mean-variance optimization,and general mean-risk optimization to formulate and study the resultingmodels. By doing so we develop a semi-parametric approach for modelingand solving general types portfolio optimization problems.

It is useful to partition the factor explained returns F^(T){tilde over(V)} into a demeaned part F^(T){tilde over (V)}₀ and its mean vectorμ=F^(T)E{tilde over (V)}, where {tilde over (V)}=μ+{tilde over (V)}₀.With this partition, the factor-explained returns are {tilde over(R)}_(F)=μ+F^(T){tilde over (V)}₀ and the factor model returns may beexpressed as {tilde over (R)}=μ+F^(T){tilde over (V)}+{tilde over (ε)}.Accordingly, we denote observed outcomes of {tilde over (V)}₀ as V_(0t)and observed outcomes of {tilde over (R)}_(F) as {tilde over (R)}_(Ft).

Expected Utility Maximization

It is desirable to use the factor model also for expected utilityoptimization. The corresponding portfolio optimization problem is statedas

max E u(1+(F ^(T) {tilde over (V)}+{tilde over (ε)})^(T) x)

Ax=b,l≤x≤h

We note that the objective function includes the expectation over therandom vector (F^(T){tilde over (V)}) and over the continuousn-dimensional random vector {tilde over (ε)}. This is why this problemis considered to be difficult.

Expected utility maximization problems have been put forward using asample-average approximation based on historical return observations inR. C. Grinold, Mean-variance and scenario-based approaches to portfolioselection, Journal of Portfolio Management, 25(2), pages 10-22, 1999:

$\max \frac{1}{T}{\sum\limits_{t}{u( {I + {R_{t}^{T}x}} )}}$Ax = b, l ≤ x ≤ h

TABLE 1 Discrete representation of the unit normal distribution z_(v)z_(v) p_(v) −5.0000 5.0000 0.000000536685270 −4.8000 4.80000.000000849689232 −4.6000 4.6000 0.000001845241790 −4.4000 4.40000.000004525164464 −4.2000 4.2000 0.000010663489269 −4.0000 4.00000.000028325893333 −3.8000 3.8000 0.000061546829887 −3.6000 3.60000.000124946943306 −3.4000 3.4000 0.000246546756629 −3.2000 3.20000.000473614054347 −3.0000 3.0000 0.000880247095147 −2.8000 2.80000.001575730929924 −2.6000 2.6000 0.002710675043685 −2.4000 2.40000.004477252136737 −2.2000 2.2000 0.007099906873363 −2.0000 2.00000.010810776888386 −1.8000 1.8000 0.015808766582307 −1.6000 1.60000.022204750725027 −1.4000 1.4000 0.029961198684615 −1.2000 1.20000.038839350894021 −1.0000 1.0000 0.048374186883117 −0.8000 0.80000.057889903396360 −0.6000 0.6000 0.066565812031542 −0.4000 0.40000.073547303686698 −0.2000 0.2000 0.078082629138417 0.0000 0.00000.079655674554058where the return observations R_(t) are calibrated to reflectforward-looking estimates of mean return and volatility. The sampleaverage model is a good approximation as long as T>>n, since only thenis the problem of full rank and statistically viable. For large-scaleutility maximization problems, arising in equities, where n>T, thesample average approximation based on historical return observations isnot a satisfactory approximation.

A more promising approach may be to use sampling from the factor modelrepresentation of asset returns. Also in this case, in order torepresent the distribution of asset returns accurately and to obtain aproblem of full rank, the sample size needs to be very large, i.e.,T>>n. However, this may be computationally prohibitive for a largenumber of assets.

Approximations to related versions of the expected utility maximizationproblem based on a factor model of asset returns have been put forwardby M. W. Brandt, P. Santa Clara, and R. Valkanov, Parametric portfoliopolicies: Exploiting characteristics in the cross section of equityreturns, Review of Financial Studies, 22(9), pages 3411-3447, 2004, andby S. De Boer, Factor tilting for expected utility maximization, Journalof Asset Management 11, pages 31-42, 2010. M. W. Brandt et al. built anexpected utility maximization model with factor exposures as thedecision variables, and the portfolio weights were subsequently derivedfrom the estimated factor loadings and the optimal factor exposures.This model assumes constant factor exposures over time. S. De Boercalculates first the expected utility optimal portfolio weights for agiven factor exposure parametrically as a function of possible factorexposures, and then solves the expected utility optimization problem inthe factor space. Both methods are, approximations and appear unable tohandle general constraints.

We proceed differently and, using the semi-parametric approach,formulate and solve the expected utility optimization problem directly:

maxΣ_(t)Σ_(v) u(1+R _(Ft) ^(T) x+σ(x)z _(w))p _(t) p _(v)

Ax=b,l≤x≤h,

where σ(x)=√{square root over (x^(T)Σx)}. This is a discrete formulationwith T*m realizations representing accurately the factor model returns,where for each outcome t there are m outcomes representing the unitnormal distribution multiplied by the nonlinear term σ(x). Like theoriginal expected utility maximization problem, this reformulatedexpected utility maximization problem is a convex problem for concaveutility functions u(·).

Gradients with respect to the decision variables x are obtained as

${\frac{\partial\;}{\partial x_{i}}{\sum\limits_{t}{\sum\limits_{v}{{u( {1 + {R_{Ft}^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}p_{v}}}}} = {\sum\limits_{t}{\sum\limits_{v}{{u^{\prime}( {1 + {R_{Ft}^{T}x} + {{\sigma (x)}z_{v}}} )}( {R_{Fi} + {\frac{1}{\sigma (x)}\sigma_{i}^{2}x_{i}z_{v}}} )p_{t}p_{v}}}}$

with σ(x)=√{square root over (x^(T)Σx)}. The expected utilitymaximization problem is then solved using a gradient-based nonlinearoptimization algorithm; see for example, MINOS, B. A. Murtagh and M. A.Saunders, Minos user's guide, Technical Report SOL 83-20, Department ofOperations Research, Stanford University, Stanford Calif. 94305, 1983.

Thus, we formulated the expected utility maximization problem as anonlinear optimization problem with linear constraints. It is a convexproblem if u(·) is concave.

Calibrating the Expected Utility Model to a Benchmark

Equilibrium returns can be obtained from the expected utilitymaximization model.

We write the vector of mean returns, μ=F^(T)E{tilde over (V)}, predictedby the factor model, as the sum of two components, an unconditionalpart, equilibrium returns implied by the market, and a conditional (onthe factor model) part:

μ=μ_(e)+μ_(e),

where μ_(e) is the vector of the unconditional equilibrium mean assetreturns and μ_(c) is the vector of the conditional part of mean assetreturns, conditioned on the factor model used. For convenience, wedefine as

{tilde over (η)}_(t) =F ^(T) {tilde over (V)} _(0t),

the demeaned factor-explained return.

Thus, the factor model returns may be written as:

{tilde over (R)}=μ _(c)+μ_(e)+{tilde over (η)}_(t)+{tilde over (ε)}

and its demeaned part as

{tilde over (R)} _(u)=μ_(e)+{tilde over (η)}_(t)+{tilde over (ε)}.

The goal in calibration is to determine an unconditional mean vectorμ_(e) such that the expected utility maximization problem

max E u(1+(μ_(e)+{tilde over (η)}_(t)+{tilde over (ε)})^(T) x)

e ^(T) x=1

for u=u_(B) results in the benchmark portfolio x₃. The benchmark weightsx_(B) are considered as representative of the weights of the marketportfolio. Thus, the equilibrium weights x_(B) imply unconditionalexpected returns μ_(e). The optimization problem includes only e^(T)x=1as a constraint, and also no bounds on holdings are needed sinceconstraints and bounds are not relevant for the benchmark portfolio.

The market equilibrium approach has been introduced for a mean-variancebased market equilibrium by F. Black and R. Litterman, Global portfoliooptimization, Financial Analysts Journal, 48(5), pages 28-43, 1992, andextended for a scenario-based utility maximization equilibrium by R. C.Grinold, Mean-variance and scenario-based approaches to portfolioselection, Journal of Portfolio Management, 25(2), pages 10-22, 1999. Wenow present an equilibrium model for expected utility optimization whenreturns follow a factor model.

The Lagrangian function is

L(x,λ)=E u(1+{tilde over (R)} _(u) ^(T) x)+λ(1−e ^(T) x)

and the optimality conditions are

$\frac{\partial{L( {x,\lambda} )}}{\partial x} = {{{{{Eu}^{\prime}( {1 + {{\overset{\sim}{R}}_{u}^{T}x}} )}R_{u}} - {\lambda \; e}} = 0}$and${\frac{\partial{L( {x,\lambda} )}}{\partial\lambda} = {{1 - {e^{T}x}} = 0}},$

where {tilde over (R)}_(u)=μ_(e)+{tilde over (η)}_(t)+{tilde over (ε)}.

For x=x_(B) and u=u_(B) (say, for the power utility function, γ=γ_(B))

E u′ _(B)(1+_(u) ^(T) x _(B)){tilde over (R)} _(u) −λe=0

Multiplying by x_(B) ^(T) on the left, one obtains

E u′ _(B)(1+{tilde over (R)} _(u) ^(T) x _(B)){tilde over (R)}_(uB)−λ=0,

by setting {tilde over (R)}_(uB)=x^(T)D{tilde over (R)}_(u), theunconditional return of the benchmark, and noting that x_(B) ^(T)e=1.Thus,

λ=E u′ _(B)(1+{tilde over (R)} _(uB)){tilde over (R)} _(uB),

Substituting for λ, one obtains for each i=1, . . . , n:

E u′ _(B)(1+{tilde over (R)} _(uB)){tilde over (R)} _(u,i) −E u′_(B)(1+{tilde over (R)} _(uB) B){tilde over (R)} _(uB)=0.

Expanding {tilde over (R)}_(u), denoting μ_(B)=μ_(e) ^(T)x_(B), {tildeover (η)}_(Bt)={tilde over (η)}_(t) ^(T)x_(B), and {tilde over(ε)}_(B)={tilde over (ε)}^(T)x_(B), one obtains for i=1 . . . , n:

E u′ _(B)(1+{tilde over (R)} _(uB))μ_(ei) +E u′ _(B)(1+{tilde over (R)}_(uB))({tilde over (η)}_(ti)+{tilde over (ε)}_(i))−E u′ _(B)(1+{tildeover (R)} _(uB))μ_(B) −Eu′ _(B)(1+{tilde over (R)} _(uB))({tilde over(η)}_(Bt)+{tilde over (ε)}_(B))=0.

Thus, one obtains for a given value of μ_(B):

${{\mu_{ei} - \mu_{B}} = {- \frac{E\; {u_{B}^{\prime}( {1 + {\overset{\sim}{R}}_{uB}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} - {\overset{\sim}{\epsilon}}_{i} - {\overset{\sim}{\epsilon}}_{B}} )}{E\; {u_{B}^{\prime}( {1 + {\overset{\sim}{R}}_{uB}} )}}}},{i = 1},\ldots \;,{n.}$

Further expanding for {tilde over (R)}_(uB)=μ_(Bt)+{tilde over(η)}_(Bt)+{tilde over (ε)}_(B), we obtain as the solution

${{\overset{.}{\mu}}_{ei} = {\mu_{B} - \frac{E\; {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\overset{\sim}{\epsilon}}_{B}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} - {\overset{\sim}{\epsilon}}_{i} - {\overset{\sim}{\epsilon}}_{B}} )}{E\; {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\overset{\sim}{\epsilon}}_{B}} )}}}},{i = 1},\ldots \;,{n.}$

One can easily see that the obtained ratio expression defines thecovariance between a random variable ⊖,

${\ominus {= \frac{- {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\overset{\sim}{\epsilon}}_{B}} )}}{E\; {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\overset{\sim}{\epsilon}}_{B}} )}}}},$

and the difference

{tilde over (R)} _(ui) −{tilde over (R)} _(uB).

Thus, one may write

μ_(ei)=μ_(B)+cov(⊖,{tilde over (R)} _(i) −{tilde over (R)} _(B))

and note that the unconditional mean return for each asset equals thebenchmark return plus the covariance between the variable ⊖, definedonly by quantities of the benchmark, and the difference between thereturn of each asset i and the benchmark return; see R. C. Grinold,Mean-variance and scenario-based approaches to portfolio selection,Journal of Portfolio Management, 25(2), pages 10-22, 1999. This reflectsan equilibrium pricing equation for expected utility optimization, andin particular here for asset returns represented by a factor model.

However, the equilibrium pricing equation did not help in the actualcomputation. Therefore we proceed to integrate the original equation andnote that the random variables {tilde over (ε)}_(t) and {tilde over(ε)}_(B) are correlated (per the definition {tilde over (ε)}_(B)={tildeover (ε)}^(T)x_(B)) with covariance

cov({tilde over (ε)}_(i),{tilde over (ε)}_(B))=x _(Bi)σ_(i) ².

Thus, one may write for i=1, . . . , n,

$\mu_{ei} = {\mu_{B} - \frac{E\; {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{1}}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} - {\sigma_{i}z_{2}} - {\sigma_{B}z_{1}}} )}{E\; {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z}} )}}}$

where σ_(B)=√{square root over (x_(B) ^(T)Σx_(B))} and where z₁ and z₂are each N(0,1) with covariance

$\frac{x_{Bi}\sigma_{i}}{\sigma_{B}}.$

With

$c_{iB} = {{{corr}( {{\overset{\sim}{\epsilon}}_{i},{\overset{\sim}{\epsilon}}_{B}} )} = \frac{ϰ\; B_{i}\sigma_{i}}{\sigma_{B}}}$

we may compute the expectation as follows:

${E\mspace{14mu} {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{1}}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} + {\sigma_{i}z_{2}} - {\sigma_{B}z_{1}}} )} = {\sum\limits_{t}\; {p_{t}{\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{{u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{1}}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} + {\sigma_{i}z_{2}} - {\sigma_{B}z_{1}}} ){p_{i}( {z_{1},z_{2}} )}{dz}_{1}{dz}_{2}}}}}}$

where p_(i)(v, w) is the density function of the bivariate unit normaldistribution,

$\mspace{20mu} {{{p( {z_{1},z_{2}} )} = {\frac{1}{2\pi \sqrt{1 - c_{iB}^{2}}}{\exp ( {- {\frac{1}{2( {1 - c_{iB}^{2}} )}\lbrack {z_{1}^{2} + z_{2}^{2} - {2c_{iB}z_{1}z_{2}}} \rbrack}} )}}},\mspace{20mu} {and}}$${E\mspace{14mu} {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z}} )}} = {\sum\limits_{t}\; {p_{t}\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{+ \infty}{{u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z}} )}e^{- \frac{z^{2}}{2}}{dz}}}}}$

where we integrate numerically between, say, −5 and 5, which we foundsufficiently accurate, using the known trapezoidal method. This had beenproposed by G. Infanger, U.S. Pat. No. 8,548,890 B2, issued Oct. 1,2013.

In the spirit of using a discrete approximation of the bivariate unitnormal distribution, one could calculate p(z_(1v) ₁ , z_(2v) ₂ )corresponding to z_(1v) ₁ , z_(2v) ₂ for given correlation coefficientcis, using numerical integration. But the probability mass functionwould have to be computed separately for each asset i due to thedependency on c_(iB). From a computational perspective, this would besimilar to doing the integration directly as described above.

Therefore, we propose the following way of integrating via discreteapproximation. Let L be the Cholesky factorization of the covariancematrix of the bivariate unit normal distribution with correlation c,i.e.,

${LL}^{T} = {\begin{pmatrix}1 & c \\c & 1\end{pmatrix}.}$

One obtains, using algebra,

$L = \begin{pmatrix}1 & 0 \\c & {\sqrt{1 - c^{2}}.}\end{pmatrix}$

One may now obtain two dependent unit normal random variables

$\begin{pmatrix}1 & 0 \\c & {\sqrt{1 - c^{2}}.}\end{pmatrix}\begin{pmatrix}z_{1} \\z_{2}\end{pmatrix}$

that is, z₁ and cz₁+√{square root over (1−c²)}z₂ as linear functions ofz₁ and z₂ that are correlated with correlation coefficient c. Thus, onemay calculate the expectations as

${E\mspace{14mu} {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{1}}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} + {\sigma_{i}z_{2}} - {\sigma_{B}z_{1}}} )} = {\sum\limits_{t}\; {\sum\limits_{v_{1}}\; {\sum\limits_{v_{2}}\; {{u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{v_{1}}}} )}( {{\overset{\sim}{\eta}}_{ti} - {\overset{\sim}{\eta}}_{Bt} + {\sigma_{i}( {{c_{iB}z_{v_{1}}} + {\sqrt{1 - c_{iB}^{2}}z_{v_{2}}}} )} - {\sigma_{B}z_{v_{1}}}} )p_{t}p_{v_{1}}p_{v_{2}}}}}}$  and${E\mspace{14mu} {u_{B}^{\prime}( {1 + \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{v}}} )}} = {\sum\limits_{t}\; {\sum\limits_{v}\; {{u_{B}^{\prime}( {1 + \; \mu_{B} + {\overset{\sim}{\eta}}_{Bt} + {\sigma_{B}z_{v}}} )}p_{t}p_{v}}}}$

by using discrete approximations (z_(v) ₁ p_(v) ₁ ), (z_(v) ₂ p_(v) ₂ ),and (z_(v),p_(v)) of the unit normal distributions, z₁, x₂, and z,respectively. The multiple sums may be readily implemented in a modelinglanguage.

For the calibration, one needs to first quantify μ_(E). Noting that{tilde over (η)}_(Bt)=(F^(T)V_(0t))^(T)x_(B) and using the benchmarkutility function u_(B) (for, say, the power utility function, γ=γ_(B)),one obtains the unconditional means μ_(e).

The calibrated expected utility maximization model

Having obtained μ_(e) from the above calibration, one may calculate

μ_(c)=μ−μ_(e)

and can write the expected utility maximization portfolio optimizationmodel as follows:

$\max \; {{Eu}( {1 + {( {{\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e} + {\overset{\sim}{\eta}}_{t} + \overset{\sim}{\epsilon}} )^{T}x}} )}$Ax = b, l ≤ x ≤ h,

where γ_(c) scales the conditional expected returns. We call γ_(c) theactive risk aversion, or the tilt parameter. The model will for u=u_(B)(for the power utility function, γ=γ_(B)) and γ_(c)→∝ result in thebenchmark portfolio if the side constraints are relaxed. For smallervalues of γ_(c) the model will tilt away from the benchmark portfolio tofollow the active predictions μ_(c). But also the overall, risk aversionγ, and more generally the utility function, may be chosen differently toobtain a suitable portfolio, different from the benchmark portfolio.

We implement the model as

$\max \mspace{14mu} \Sigma_{t}\Sigma_{v}\mspace{11mu} {u( {1 + {( {{\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e} + {F^{T}V_{0t}}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}p_{v}$${{Ax} = b},{l \leq x \leq h},{{{where}\mspace{14mu} {\sigma (x)}} = {\sqrt{x^{T}{\Sigma x}}.}}$

The result is a powerful model for active portfolio management, with thepotential to effectively control downside risk by using an appropriatechoice of utility function. It can be implemented in a modelinglanguage, since the expected value is calculated based on multiple sumsof discrete realizations representing with sufficient accuracy thefactor model returns. No functions carrying out numerical integrationneed to be programmed.

Mean-Variance Optimization

In practical implementations of mean-variance portfolio optimization,the mean vector and the covariance matrix M need to be estimated.Historical, observations R_(t), t=1, . . . , T, of {tilde over (R)} maybe used to estimate the quantities. However, for a large number ofassets, i.e., n>T, using sample averages directly to estimate the meanvector and the covariance matrix do not give the desired results,because the sample errors tend to be large and also the resultingcovariance matrix is rank deficient (positive semidefinite rather thanpositive definite). In order to overcome this problem, factor models, asdiscussed above have been applied to model asset returns.

Using the factor model representation, the covariance matrix can berepresented as

M=F ^(T) M _({tilde over (V)}) F+Σ,

where M_({tilde over (V)}) is the k×k covariance matrix of the factors(or factor returns), and Σ=diag(σ_(i) ²) is the diagonal matrix ofidiosyncratic variance, where σ_(i) ² is the variance of the i-thindependent error term {tilde over (ε)}₂. The matrixM_({tilde over (V)}) is, typically estimated using historicalobservations. Presuming all asset returns have positive variance,because Σ is diagonal, the resulting covariance matrix M is of full rank(rank(M)=n). The number of parameters to be estimated (nk for the factorloadings+k(k+1)/2 for the factor covariances+k for the means) is muchsmaller than without imposing the linear factor model ((n(n+1)/2 for thecovariances+n for the means), especially when the number of factors k iskept reasonably small.

The mean-variance portfolio optimization problem based on a factor modelrepresentation of asset returns is

${\max \; {E( {F^{T}\overset{\sim}{V}} )}^{T}x} - {\frac{\gamma}{2}{x^{T}( {{F^{T}M_{\overset{\sim}{V}}F} + \sum} )}x}$Ax = b, l ≤ x ≤ h

An equivalent formulation arises when using the semi-parametricformulation of the factor model in accordance with the present inventiondirectly:

${\max \; \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{( {{( {F^{T}V_{0\; t}} )^{T}x} + {{\sigma (x)}u_{v}}} )^{2}p_{t}p_{v}}}}}$Ax = b, l ≤ x ≤ h

where σ(x)=√{square root over (x^(T)Σx)}. This is a scenario formulationof the mean-variance problem with Tm scenarios representing thecovariance structure. The number of data points m representing N(0,1)determines the accuracy of the formulation. There does not appear to bea particular advantage of the scenario formulation over the typicalfactor model mean-variance formulation. If the unit normal distributionrepresentation is sufficiently accurate, the scenario formulation willresult in the same optimal portfolios as the typical factor modelmean-variance formulation.

Calibrating the Mean-Variance Model to a Benchmark

The mean-variance portfolio optimization problem for calibratingunconditional expected returns is

${\max \mspace{14mu} \mu_{e}^{T}x} - {\frac{\gamma_{B}}{2}x^{T}M\; x}$e^(T)x = 1

where μ_(e) is the n-vector of unconditional mean returns to bedetermined, M=F^(T)M_({tilde over (V)})F+Σ is the n×n covariance matrixof asset returns, and γ_(B) is the risk aversion parameter of thebenchmark.

The Lagrangian function of the mean-variance problem is

${L( {x,\lambda} )} = {{\mu_{e}^{T}x} - {\frac{\gamma_{B}}{2}x^{T}M\; x} + {{\lambda ( {1 - {e^{T}x}} )}.}}$

Setting all derivatives to zero one obtains:

$\frac{\partial{L( {x,\lambda} )}}{\partial x} = {{\mu_{e} - {\gamma_{B}M\; x} - {e\; \lambda}} = 0}$and$\frac{\partial{L( {x,\lambda} )}}{\partial\lambda} = {{1 - {e^{T}x}} = 0.}$

Multiplying on the left by x^(T) _(B), one obtains

x _(B) ^(T)μ_(e)−γ_(B) x _(B) ^(T)(Mx _(B))−λ=0

and it follows that

λ=μ_(B)−γ_(B)σ₂ ^(B),

where μ_(B)=μ_(e) ^(T)x_(B) and σ_(B) ²=x_(B) ^(T)x_(B). Now we obtainfor a given value of γ_(B)

μ_(e)=γ_(B) Mx _(B)+(μ_(B)−γ_(B)σ_(B) ²)e

Note that the second term (μ_(B)−γ_(B)σ_(B) ²)e is a constant and whenadded to the objective does not affect the solution of the mean-varianceportfolio optimization problem. Therefore that term may be dropped.Thus,

μ_(e)=γ_(B) Mx _(B).

We actually calculate the equilibrium returns μ_(e) by exploiting thefactor model form of the covariance matrix as

μ_(e)=γ_(B)(F ^(T) M _({tilde over (V)}) F+Σ)x _(B).

Equilibrium returns obtained for the mean-variance model, of course,differ from those obtained earlier for expected utility optimization,but both are labeled as μ_(e). It should be clear from the context whichis referred to.

The calibrated mean-variance model

Having obtained μ_(e) from the calibration, we calculate

μ_(c)=μ−μ_(e)

and may now write the mean-variance portfolio optimization model as:

${\max \mspace{14mu} ( {{\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e}} )^{T}x} - {\frac{\gamma}{2}{x^{T}( {{F^{T}M_{\overset{\sim}{V}}F} + \Sigma} )}x}$Ax = b, l ≤ x ≤ h

or equivalently, using the semi-parametric formulation, as

${{{{\max ( {{\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e}} )}^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}^{\;}\; {\sum\limits_{v}^{\;}{( {{( {F^{T}V_{0\; t}} )^{T}x} + {{\sigma (x)}z_{v}}} )^{2}p_{t}p_{v}{Ax}}}}}} = b},{l \leq x \leq h}$

The model will for γ=γ_(B) and γ_(c)→∞ result in the benchmark portfolioif the side constraints are relaxed. For smaller values of γ_(c) themodel will tilt away from the benchmark portfolio to follow the activepredictions μ_(c). The overall risk aversion γ may be chosen differentlyto obtain a suitable portfolio with a different risk than that of thebenchmark portfolio.

Mean-Risk Optimization

Mean-risk optimization typically concerns portfolio optimizationproblems with risk measures other than portfolio variance, whilemean-variance optimization is also a subset of mean-risk optimizationwith variance as the risk measure. Risk measures under consideration areeither dispersion measures or downside risk measures. Besides variance(or standard deviation), important dispersion measures are mean absolutedeviation and mean absolute moments. Important downside measures aresemi-variance and lower partial moments, but most important in financeare the tail measures Value-at-Risk (VaR) and Conditional-Value-at-Risk(CVaR). We will present the formulation of the latter first and thendiscuss the other risk measures considered in mean-risk optimization.

Value-at-Risk and Conditional Value-at-Risk

Value-at-Risk is defined as

VaR_(α)({tilde over (R)} ^(T) x)=min{W: P(−{tilde over (R)} ^(T)x>W)≤α)}.

It is the smallest number W such that the probability of a loss greaterthan W is no more than α. The quantity α is called the loss tolerance,whereas the quantity (1−α) is referred to as the confidence, level.VaR_(α) defines a quantile, for example the 5% quantile when α=0.05. Anequivalent definition is VaR_(α)({tilde over (R)}^(T)x)=−max{W: P({tildeover (R)}^(T)x<W)≤α)}, where Value-at-Risk Is defined as the negative ofthe largest value W such that the probability of a return less than W isno more than α. The negative sign reflects that VaR is defined as a lossand not as a return.

For general distributions, Value-at-Risk is not a coherent measure ofrisk. It is not convex with respect to the portfolio variables x and istherefore difficult to optimize. However, Conditional-Value-at-Risk,defined as

CVaR_(α)({tilde over (R)} ^(T) x)=E{−{tilde over (R)} ^(T) x|{tilde over(R)} ^(T) x≤−VaR_(α)({tilde over (R)} ^(T) x)},

is convex with respect to the portfolio variables x. It is thereforewell suited to mean-risk optimization, when tail risk is to becontrolled. CVaR_(α) is the expected value of losses greater than orequal to the VaR_(α) quantile.

Other names for Conditional Value-at-Risk are Expected Shortfall andTail Value of Risk. Since the losses in the tail are at least as much asVaR_(α)({tilde over (R)}^(T)x), and CVaR_(α)({tilde over (R)}^(T)x) isthe average of these, it follows that

CVaR_(α)({tilde over (R)} ^(T) x)≥VaR_(α)({tilde over (R)} ^(T) x).

Thus, CVaR_(α) always dominates VaR_(α). Therefore one may use CVaR_(α)in an optimization as a conservative approximation to VaR_(α).

Let

Ψ({tilde over (R)} ^(T) x,W)=∫_({tilde over (R)}) _(T) _(x≤−W) p({tildeover (R)})d{tilde over (R)}

be the probability of {tilde over (R)}^(T)x not exceeding a threshold−W.For a fixed portfolio x it is the cumulative distribution function of aloss associated with portfolio x. It is generally nondecreasing withrespect to −W and is continuous with respect to −W as long as there areno jumps. For simplicity, only distributions with conthmous distributionfunctions are considered. However, while involving substantial technicaldetail, the analysis extends to non-continuous distribution functions aswell.

Using the definition of Ψ, VaR_(α)({tilde over (R)}^(T)x) may be writtenas

VaR_(α)({tilde over (R)} ^(T) x)=min{W: Ψ({tilde over (R)} ^(T) x,W)≤α)}

and CVaR_(α)({tilde over (R)}^(T)x) may also be written as

${{CVaR}_{\alpha}( {{\overset{\sim}{R}}^{T}x} )} = {\frac{1}{\alpha}{\int_{{{\overset{\sim}{R}}^{T}x} \leq {- {{VaR}_{\alpha}{({{\overset{\sim}{R}}^{T}x})}}}}{{- {\overset{\sim}{R}}^{T}}{{xp}( \overset{\sim}{R} )}d\overset{\sim}{R}}}}$

Based on the above definitions; R. T. Rockafellar and S. Uryasev,Optimization of conditional value-at-risk, Portfolio Safeguard byAOrDa.com, 2(3), pages 21-41, 2000, defined the following function:

${{F_{\alpha}( {{{\overset{\sim}{R}}^{T}x},W} )} = {W + {\frac{1}{\alpha}E\{ ( {{{- {\overset{\sim}{R}}^{T}}x} - W} )^{+} \}}}},$

where F_(a)({tilde over (R)}^(T)x, W), as a function of W, is convex andcontinuously differentiable in W for any value of x. Accordingly,CVaR_(α) and VaR_(α) for any value of x may be obtained as:

${C\; V\; a\; {R_{\alpha}( {{\overset{\sim}{R}}^{T}x} )}} = {\min\limits_{W}{{F_{\alpha}( {{{\overset{\sim}{R}}^{T}x},W} )}\mspace{20mu} {and}}}$${{VaR}_{\alpha}( {{\overset{\sim}{R}}^{T}x} )} \in {\arg \; {\min \mspace{14mu}}_{W}{{F_{\alpha}( {{{\overset{\sim}{R}}^{T}x},W} )}.}}$

Furthermore, CVaR_(α)({tilde over (R)}^(T)x) is also convex. That isF_(α)({tilde over (R)}(x), W) is convex for convex functions {tilde over(R)}(x) and, therefore, also for the linear function {tilde over(R)}^(T)x. Thus, CVaR is amenable to optimization:

${{\min\limits_{x}{C\; V\; a\; {R_{\alpha}( {{\overset{\sim}{R}}^{T}x} )}}} = {\min\limits_{W,x}\; {F_{\alpha}( {{{\overset{\sim}{R}}^{T}x},W} )}}},$

where the function F_(α)({tilde over (R)}^(T)x, W) is jointly minimizedover (W, x). Given linear (or convex) portfolio constraints, one mayminimize this convex function subject to linear (or convex) constraints.Thus, the obtained CVaR portfolio optimization problem is a convexoptimization problem.

In particular, R. T. Rockafellar and S. Uryasev, Optimization ofconditional value-at-risk. Portfolio Safeguard by AOrDa.com, 2(3), pages21-41, MOO showed that when using a sample R_(ω), ω∈S_(ω) of {tilde over(R)},

${F_{\alpha}( {{R_{\omega}^{T}x},W} )} = {W + {\frac{1}{\alpha}\frac{1}{S_{\omega}}{\sum\limits_{\omega}( {{{- {\overset{\sim}{R}}_{\omega}^{T}}x} - W} )^{+}}}}$

and, for this case, the portfolio optimization problem can be formulatedas a linear program:

$\begin{matrix}{{{{\min \mspace{11mu} W} + {\frac{1}{\alpha}\frac{1}{S_{\omega}}{\sum\limits_{\omega}v_{\omega}}}} =}} & {{CVaR}_{\alpha}} \\{{{v_{\omega} + {R_{\omega}^{T}x_{i}} + W} \geq 0},} & {{{v_{\omega} \geq 0},}} \\{{{e^{T}x} =}} & {1} \\{{{\mu^{T}x} \geq}} & {{\overset{\_}{r}}_{P}}\end{matrix}$

where {tilde over (r)}_(p) is a predefined desired value of portfolioexpected return. At the optimal solution one obtains W*=VaR_(α)(R_(ω)^(T),x*), the Value-at-Risk of the optimal portfolio. Note thatVaR_(α)(R_(ω) ^(T)x*) is a sample average approximation ofVaR_(α)({tilde over (R)}^(T)x*) based on a return sample ω∈Sω_(.)

We proceed differently by using the factor model returns in theexpression for CVaR_(α)({tilde over (R)}^(T)x)=minus F_(α)({tilde over(R)}^(T)x, W) to obtain:

${{CVaR}_{\alpha}( {{\overset{\sim}{R}}^{T}x} )} = {\min\limits_{W}\{ {W + {\frac{1}{\alpha}E\{ ( {{{- ( {{F^{T}\overset{\sim}{V}} + \overset{\sim}{ɛ}} )^{T}}x} - W} )^{+} \}}} }$

which is a convex function in x. Then,

${{CVaR}_{\alpha}( {R^{T}x} )} = {\min\limits_{W}\{ {W + {\frac{1}{\alpha}E\{ ( {{{- ( {F^{T}\overset{\sim}{V}} )^{T}}x} + {{\sigma (x)}z} - W} )^{+} \}}} }$

is a convex function in x. Using the semi-parametric factor modelrepresentation of asset returns in accordance with the present inventionwe calculate the expectation by multiple sums to obtain

${{CVaR}_{\alpha}( {R_{t\; v}^{T}x} )} = {\min\limits_{W}\{ {{W + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{\{ ( {{{- ( {F^{T}V_{t}} )^{T}}x} + {{\sigma (x)}z_{v}} - W} )^{+} \} p_{t}p_{v}}}}}},} }$

which is convex in x. One may implement this function via a nonlinearprogramming formulation as:

${\min \mspace{14mu} W} + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{\upsilon}{v_{t\; v}p_{t}p_{v}}}}}$s/t  (F^(T)V_(t))^(T)x + σ(x)z_(v) + v_(t v) + W ≥ 0,  v_(t v) ≥ 0, ∀t, v

and minimize it subject to linear or convex portfolio constraints,thereby solving a convex program.

Changing the sign of the objective and maximizing an afflne transform ofthe objective with a positive coefficient leads to maximizing a concavefunctiOn subject to linear (or convex) constraints; a convexoptimization problem. The following presents a new formulation of themean-risk portfolio optimization probleth with CVaR as the risk measurefor the factor model representation R^(T)x=(F^(T)V_(t)+{tilde over(ε)})^(T)x of asset returns as

${\max \mspace{14mu} \mu^{T}x}\mspace{31mu} - {\frac{\gamma}{2}( {W + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{v_{t\; v}p_{t}p_{v}}}}}} )}$$\begin{matrix}{( {F^{T}V_{t}} )^{T}x} & {{{+ {\sigma (x)}}z_{v}} + \nu_{t\; v} + W} & {{{\geq 0},{v_{t\; v} \geq 0},\mspace{14mu} {\forall t},v}\mspace{11mu}} \\{Ax} & \; & {{= b},\; {l \leq x \leq h}}\end{matrix}$

where σ(x)=√{square root over (x^(T)Σx)}. At the optimal solution,

${{W^{*} + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{v_{t\; v}^{*}p_{t}p_{v}}}}}} = {{CVaR}_{\alpha}( {{\overset{\_}{R}}^{T}x^{*}} )}},$

the optimal Conditional Value-at-Risk value, and W*=VaR_(α)({tilde over(R)}^(T)x*), the Value-at-Risk value of the returns of the optimalportfolio.

Note that by using the semi-parametric factor model representation ofasset returns in accordance with the present invention a nonlinearprogram based on the nonlinear discrete formulation with Tm realizationsis obtained. It is a convex problem as outlined above. We solve theproblem by using gradient-based nonlinear programming techniques.

As in the case of mean-variance portfolio optimization, one would use

${\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e}$

instead of μ, in order to facilitate a measured approach to activeportfolio management. The equilibrium returns μ_(e) would be calibratedusing the mean-variance model. The same also applies to the followingmodels entailing different risk measures.

Other Risk Measures

Other Dispersion Measures

The following describes the mean-absolute-deviation (MAD) and themean-absolute-moment (MAM) measures as part of mean-risk optimization.

The mean-absolute-deviation measure,

Risk({tilde over (R)} ^(T) x)=MAD({tilde over (R)} ^(T) x)=E|{tilde over(R)} ^(T) x−E({tilde over (R)} ^(T) x)|,

introduced by H. Konno and H. Yamazaki, Mean-absolute deviationportfolio optimization model and its application to Tokyo stock market,Management Science, 37(5), pages 519-531. 1991, results in exactly thesame optimal portfolio as mean-variance optimization if asset returnsare multi-variate normally distributed. However, the portfolios aredifferent to the extent that asset returns deviate froth themulti-variate normal distribution. When a sample of asset returns isused to represent the asset distribution, the mean-MAD model results ina linear program, and is, thus, easier to solve than the quadraticmean-variance problem. However, using factor models of asset returns,large-scale mean-variance models are routinely solved in a short time.

We now introduce a new formulation of the mean-MAD model for assetreturns following the factor model R^(T)x=(F^(T){tilde over (V)}+{tildeover (ε)})^(T)x using the semi-parametric approximation:

${\max \mspace{14mu} \mu^{T}x}\mspace{14mu} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{( {v_{t\; v}^{+} + \nu_{t\; v}^{-}} )p_{t}p_{v}}}}}$$\begin{matrix}{{- ( {F^{T}V_{0\; t}} )^{T}}x} & {{{- {\sigma (x)}}z_{v}} + \nu_{t\; v}^{+} - \nu_{t\; v}^{-}} & {{{= 0},v_{t\; v}^{+},{\nu_{t\; v}^{-} \geq 0},\mspace{14mu} {\forall t},v}\mspace{11mu}} \\{Ax} & \; & {{= b},\; {l \leq x \leq h}}\end{matrix}$

where σ(x)=√{square root over (x^(T)Σx)}. It is a convex nonlinearproblem, and can be solved using a modern nonlinear programming solver.

The mean-absolute-moment measure is defined as

Risk({tilde over (R)} ^(T) x)=MAM _(q)({tilde over (R)} ^(T) x)=E|{tildeover (R)} ^(T) x−E({tilde over (R)} ^(T) x)|^(q) ,q>1,

Using the semi-parametric factor model representation of asset returns,the mean-MAM model can be stated directly as a convex nonlinear program(for q>1):

${\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{\lbrack ( {{( {F^{T}V_{0\; t}} )^{T}x} + {{\sigma (x)}z_{v}}} )^{2} \rbrack^{q/2}p_{t}p_{v}}}}}$Ax = b, l ≤ x ≤ h

where σ(x)=√{square root over (x^(T)Σx)}, and where V_(0t) is theobserved demeaned part of the factor return. In order to account for theabsolute value, for odd q, we first square the term in the objective andthen raise it to the power of q/2. The problem may be solved using agradient-based nonlinear programming solver. Note that for q=1,MAM₁({tilde over (R)}^(T)=MAD({tilde over (R)}^(T)x). Thus, themean-MAM_(I) portfolio optimization problem equals the mean-MAD modeldescribed above.

An equivalent formulation that does not rely on the “squaring and thenraising to the power of q/2” method of calculating themean-absolute-moment is:

${\max \mspace{14mu} \mu^{T}x}\mspace{14mu} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{( {v_{t\; v}^{+} + \nu_{t\; v}^{-}} )^{q}p_{t}p_{v}}}}}$$\begin{matrix}{{- ( {F^{T}V_{0\; t}} )^{T}}x} & {{{- {\sigma (x)}}z_{v}} + \nu_{t\; v}^{+} - \nu_{t\; v}^{-}} & {{{= 0},v_{t\; \nu}^{+},{\nu_{t\; \nu}^{-} \geq 0},\mspace{14mu} {\forall t},v}\mspace{11mu}} \\{Ax} & \; & {{= b},\; {l \leq x \leq h}}\end{matrix}$

It is a convex problem, and can be solved using a gradient-basednonlinear programming solver.

Other Downside Measures

The following describes the semi-variance and the lower-partial-momentmeasures as part of mean-risk optimization.

The semi-variance is defined as,

Risk({tilde over (R)} ^(T) x)=σ_(semi) ²({tilde over (R)} ^(T)z)=Emin({tilde over (R)} ^(T) x−E({tilde over (R)} ^(T) x),0)²,

where only return outcomes smaller than the expected return areconsidered in the standard deviation. Squaring makes the negative returnvalues positive. Thus, the semi-variance, as defined above, is a riskmeasure in the sense that negative returns represent risk. Using thesemi-parametric factor model representation of asset return, themean-semi-variance model can be stated as:

${\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{v_{t\; v}^{2}p_{t}p_{v}}}}}$$\begin{matrix}{{( {F^{T}V_{0\; t}} )^{T}x} + {{\sigma (x)}z_{v}} + \nu_{t\; v}} & {{{= 0},{v_{t\; \nu} \geq 0},\mspace{14mu} {\forall t},v}\mspace{11mu}} \\{Ax} & {{= b},\; {l \leq x \leq h}}\end{matrix}$

where σ(x)=√{square root over (x^(T)Σx)}, and where V_(0t) is thedemeaned part of the factor return. It is a convex nonlinear program,and is solved using a gradient-based nonlinear programming solver.

Lower partial moments are defined as

Risk({tilde over (R)} ^(T) x)=LPM _(q) w({tilde over (R)} ^(T)x)=E(−min({tilde over (R)} ^(T) x−W,0))^(q) ,q≥1

where W is a predefined value of return and risk is considered as theexpected value of the negative of returns that are below the level Wraised to the power of q. Using the semi-parametric factor modelrepresentation of asset returns, the mean-LPM model can be stated as:

${\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{v_{t\; v}^{q}p_{t}p_{v}}}}}$$\begin{matrix}{{( {F^{T}V_{\; t}} )^{T}x} + {{\sigma (x)}z_{v}} + \nu_{t\; v}} & {{{\geq W},{v_{t\; v} \geq 0},\mspace{14mu} {\forall t},v}\mspace{11mu}} \\{Ax} & {{= b},\; {l \leq x \leq h}}\end{matrix}$

where σ(x)=√{square root over (x^(T)Σx)} and where W is a predefinedconstant. For q=1, risk is a linear measure, namely, the expected valueof returns below the level W. In this case, the mean-LPM model becomes alinear program: For q>1 the mean-LPM model is a convex nonlinearprogram, and may be solved using a nonlinear programming solver.

Risk Constraints

Constraints on risk measures may be effectively used to control certainrisk metrics in a mean-variance optimization or expected utilityoptimization. For example, one may want to explicitly control varianceor tracking error in an expected utility problem, or one may want tocontrol Value-at-Risk in a mean-variance problem. With the differentrisk measures discussed above, and considering mean-variance, expectedutility, and mean-risk optimization problems, there are manycombinations to analyze. For example, one combination is controllingValue-at-Risk in a mean-variance optimization problem by adding, aConditional Value-at-Risk constraint. We use the semi-parametricrepresentation of the factor model returns in the problem formulation.Let p be a given maximal Value-at-Risk level. One may formulate the CVaRconstraint as part of the mean-variance optimization, as follows:

${\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{x^{T}( {{F^{T}M_{\overset{\_}{V}}F} + \sum} )}x}$Ax = b, l ≤ x ≤ h${{W + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}p_{t}p_{v}}}}}} \leq {{{\rho ( {F^{T}V_{t}} )}^{T}x} + {{\sigma (x)}z_{v}} + u_{tv} + W} \geq 0},{u_{tv} \geq 0},{\forall t},v$

If the CVaR constraint is binding in the optimal solution, the variableW* represents the VaR value and the expression

$W^{*} + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{*}p_{t}p_{v}}}}}$

represents the CVaR value of the optimal solution x* of the problem. Ifthe CVaR constraint is not binding, the obtained values of W* and

$W^{*} + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{*}p_{t}p_{v}}}}}$

may be arbitrary, and one only knows that both VaR and CVaR at theoptimal solution x* are strictly less than ρ. A minor modification ofthe problem, where one assigns an extra variable to CVaR and penalizesit ever so slightly in the objective, will also provide VaR and CVaRvalues, when at the optimal solution the CVaR constraint is not binding.

Since the CVaR constraint is a convex constraint, the problem is aconvex nonlinear programming problem.

Moment Constraints

As part of a portfolio optimization problem, higher moments of thereturns distribution, in particular, skewness and/or kurtosis ofportfolio returns, may need to be constrained. With portfolio returnsdefined as {tilde over (R)}^(T)x, for a given portfolio x, skewness isdefined as

${Skew} = \frac{{E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{3}}{( {E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{2} )^{\frac{3}{2}}}$

and kurtosis is defined as the fourth standardized moment,

${Kurt} = \frac{{E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{4}}{( {E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{2} )^{2}}$

The term {tilde over (R)}^(T)x−E({tilde over (R)}^(T)x) in the abovedefinitions represents the demeaned portfolio returns. The normaldistribution has a skewness of 0 and a kurtosis of 3. Therefore, excesskurtosis (K−3) is often used instead of kurtosis. Leptokurticdistributions (with a peakedness higher than that of the normaldistribution) tend to have long tails also. An investor may seek areturns distribution that is not too negatively skewed and has limitedkurtosis, in order to limit heavy left tails. Neither skewness norkurtosis are convex functions of x.

We use the semi-parametric representation of factor model returns andcalculate skewness and kurtosis. Using a variable u_(tv) representingthe demeaned portfolio returns,

v _(tv)=(F ^(T) V _(0t))^(T) x+σ(x)z _(v),

where σ(x)=√{square root over (x^(T)Σx)}, skewness is expressed as

${Skew} = \frac{{Eu}_{tv}^{3}}{( {Eu}_{tv}^{2} )^{\frac{3}{2}}}$

and kurtosis as

${Kurt} = {\frac{{Eu}_{tv}^{4}}{( {Eu}_{tv}^{2} )^{2}}.}$

Constraining skewness to be greater than or equal to a given lower boundSkews, we formulate

Ev _(tv) ³−Skew_(l)(Ev _(tv) ²)^(3/2)≥0,

and using summation for calculating the expectations based on thediscrete formulation of the factor model returns, one obtains

${{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{3}p_{t}p_{v}}}} - {{Skew}_{l}( {\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{2}p_{t}p_{v}}}} )}^{\frac{3}{2}}} \geq 0.$

The formulas simplify if one seeks to constrain skewness to benonnegative (Skew≥0),

${\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{3}p_{t}p_{v}}}} \geq 0.$

Constraining kurtosis to be less than or equal to a given upper boundKurt_(h) yields the following relation

Ev _(tv) ⁴−Kurt_(h)(Ev _(tv) ²)²≤0

and using summation for calculating the expectations based on thediscrete formulation of the factor model returns one obtains

${{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{4}p_{t}p_{v}}}} - {{Kurt}_{h}( {\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{2}p_{t}p_{v}}}} )}^{2}} \leq 0.$

Estimates based on samples of ratios of expectations and expectationstaken to a power may be biased. Other formulations for sample skewnessand sample kurtosis exist that include factors for bias correction.

As an example, a mean-variance model may be formulated where skewness isconstrained to be nonnegative and kurtosis is constrained to be lessthan or equal to three:

${{\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{2}p_{t}p_{v}{{Ax} = b}}}}}},{{l \leq x \leq {h - {( {F^{T}V_{0\; t}} )^{T}x} - {{\sigma (x)}z_{v}} + u_{tv}}} = 0}$${\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{3}p_{t}p_{v}}}} \geq 0$${{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{4}p_{t}p_{v}}}} - {3( {\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{2}p_{t}p_{v}}}} )^{2}}} \leq 0.$

where σ(x)=√{square root over (x^(T)Σx)}. It is a non-convexoptimization problem, and multiple local optima may exist.

Derivative Securities

Derivative securities are securities whose price depends on the price ofan underlying security. Derivative securities include, for example,options, fonvards, futures, swaps and others. Derivatives may beconsidered on an underlying, portfolio of assets, which includes optionson an individual asset or options on an index. In order to obtain thereturn of a derivative security, its value is determined at the end ofthe investment period depending the value (return) of the underlyingsecurity. This may be shown using the example of options on a portfolioof assets (index). Let K be the strike price and p be the price of theoption, expressed as a fraction of the price of the underlying security(index). The return of a call option is

${r^{call} = {\frac{\max \{ {0,{{p_{s\; 0}( {1 + {\overset{\sim}{R}}_{U}} )} - {p_{s\; 0}K}}} \}}{p_{s\; 0}p} - 1}},$

where p_(a0) is the price of the underlying security at the beginning ofthe investment period and {tilde over (R)}_(U) is its return. Dividingby p_(s0) one obtains:

${r^{call} = {\frac{\max \{ {0,{( {1 + {\overset{\sim}{R}}_{U}} ) - K}} \}}{p} - 1}},$

the return of a call option depending: on the return of the underlyingsecurity. Similarly, the return of a put option is

$r^{put} = {\frac{\max \{ {0,{k - ( {1 + {\overset{\sim}{R}}_{U}} )}} \}}{p} - 1}$

depending on the return of the underlying security. Let

be a specific option, put or call, with relative strike price K andrelative price p. Then, one may define the return of an option

on the underlying portfolio x_(U) as

r _(l) =f _(l)({tilde over (R)} ^(T) x _(U))

where f_(l)(·) is the return generating function of option

and {tilde over (R)}_(U)={tilde over (R)}^(T)x_(U). In general, one maydefine τ_(l)=f_(l)({tilde over (R)}^(T)x_(U)) as the return generationfunction of derivative security

, defining its return based on the return {tilde over (R)}^(T)x_(U) ofthe underlying portfolio of assets.

Let f be the n_(d)-vector of return generating functions of variousderivative securities and let y be the n_(d)-vector of holdings of thesederivative securities. In portfolio optimization, one needs to maximizethe expectation of a function G of the portfolio return. Accordingly,with derivatives in the portfolio one obtains:

max EG({tilde over (R)} ^(T) x+f ^(T)({tilde over (R)} ^(T) x _(U))y)

subject to portfolio constraints on n+n_(d) assets. Substituting thefactor model representation. we obtain:

max EG((F ^(T) V _(t)+{tilde over (ε)})^(T) x+f ^(T)((F ^(T) V_(t)+{tilde over (ε)})^(T) x _(U))y)

and using the semi-parametric reformulation described above,

max EG(R _(Ft) ^(T) x+σ(x)z ₁ +f ^(T)(R _(Ft) ^(T) x _(U)+σ_(U) z ₂)y)

where z₁ and z₂ are unit normal random variables correlated withcorrelation coefficient

$c_{xU} = {\frac{x^{T}{\sum x_{u}}}{{\sigma (x)}\sigma_{U}} = \frac{\sum{x_{i}x_{Ui}\sigma_{i}^{2}}}{\sqrt{\sum\limits_{i}^{\;}{\sigma_{i}^{2}x_{i}^{2}}}\sigma_{U}}}$

and where σ_(U)=√{square root over (x_(U) ^(T)Σx_(U))}=√{square rootover (Σ_(i)σ_(i) ²x_(Ui) ²)}. For any given value of x and y one mayintegrate this function using two-dimensional numerical integration as

$\max \; {\sum\limits_{t}{\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{{G( {{R_{Fi}^{T}x} + {{\sigma (x)}z_{1}} + {{f^{T}( {{R_{Fi}^{T}x_{U}} + {\sigma_{U}z_{2}}} )}y}} )}p_{t}{p( {z_{1},z_{2}} )}{dz}_{1}{dz}_{2}\mspace{14mu} {where}}}}}$${p( {z_{1},z_{2}} )} = {\frac{1}{2\pi \sqrt{1 - c_{xU}^{2}}}{{\exp ( {- {\frac{1}{2( {1 - c_{xU}^{2}} )}\lbrack {z_{1}^{2} + z_{2}^{2} - {2c_{iB}z_{1}z_{2}}} \rbrack}} )}.}}$

This was proposed in G. Infanger, U.S. Pat. No. 8,548,890 B2, issuedOct. 1, 2013, where the numerical integration was carried out using thetrapezoidal method integrating between −5 and 5. Gradients were providedand the resulting problem was solved using a gradient-based nonlinearoptimization algorithm. The approach worked well and led to shortcomputation times.

We now extend the approach to using the discrete approximation of theunit normal variables. Using the Cholesky factorization described above,one obtains two dependent unit random variables with correlationcoefficient c_(xU) as z₁ and c_(xU)z₁+√{square root over (1−c_(xU)²)}z₂, where now z₁ and z₂ are two independent unit normal variables.Using the discrete approximations (z_(v) ₁ ,p_(v) ₁ ) and (z_(v) ₂,p_(v) ₂ ) for z₁ and z₂, one obtains:

$\max \; {\sum\limits_{t}{\sum\limits_{v_{1}}{\sum\limits_{v_{2}}{{G( {{R_{Fi}^{T}x} + {{\sigma (x)}z_{v_{1}}} + {{f^{T}( {{R_{Fi}^{T}x_{U}} + {\sigma_{U}( {{c_{xU}z_{v_{1}}} + {\sqrt{1 - c_{xU}^{2}}z_{v_{2}}}} )}} )}y}} )}p_{t}p_{v_{1}}p_{v_{2}}}}}}$

subject to portfolio constraints. The portfolio optimization problemincluding derivatives on an underlying portfolio of assets (index) cannow be carried out using multiple sums. Thus, it can be implemented in amodeling system. Note that a single asset is a portfolio with oneposition and, thus, derivatives on a single asset are covered by thisapproach in the same way. However, derivatives on each of multipleunderlying assets cannot be handled by the approach, since this wouldlead to many multiple sums (two plus one for each underlying asset) andwould be computationally prohibitive.

Calculating Performance Statistics

In any portfolio optimization context, it is often useful to provideforward-looking (or a ante) statistics, including (predicted values of)portfolio mean return, standard deviation, and Sharpe ratio. These canbe calculated based on the covariance structure of asset returns.Forward-looking statistics involving downside measures, includingdownside standard deviation, or Value-at-Risk are more difficult tocalculate, unless they are based on assuming normally distributedreturns. Using the factor model returns for a given portfolio x,forward-looking statistics involving downside risk can be calculated,taking advantage of the semi-parametric formulation of the factor modelreturns. The following shows this for the Sortino (SoR) ratio and forValue-at-Risk (VaR).

Forward-Looking Downside Target Standard Deviation and Sortino Ratio

For a Given Portfolio x_(P). We May Write the Factor Model Returns as

R _(Ptv) =R _(Ft) ^(T) x _(P)+σ(x _(P))z _(v),

where R_(Ptv) are Tm forward-looking return realizations withcorresponding probability p_(t)p_(v), representing the forward-lookingreturns distribution of portfolio x_(P).

The downside target standard deviation is defined as

√{square root over (E[min({tilde over (R)} ^(T) x _(P) −r _(f),0)]²)}

and is the square root of the lower partial moment of order 2 of {tildeover (R)}^(T)x_(P), with the risk-free rate r_(f) as the target rate.

The Sortino Ratio is defined as

${SoR} = {\sqrt{n_{Y}}\frac{E( {{{\overset{\sim}{R}}^{T}x_{P}} - r_{f}} )}{\sqrt{{E( {\min ( {{{{\overset{\sim}{R}}^{T}x_{P}} - r_{f}},0} )} \rbrack}^{2}}}}$

This is the scaled ratio of the excess returns (over the risk-free rater_(f)) divided by the downside target standard deviation. It istypically expressed in annual terms. If returns are observed n_(Y) timesper year, the Sortino ratio obtained from the observations is annualizedby multiplying with √{square root over (n_(Y))}. For example, when usingmonthly observations, one multiplies by √{square root over (12)}.

The forward-looking Sortino Ratio based on the semi-parametric factormodel returns is

${SoR} = {\sqrt{n_{Y}}\frac{{E_{t}R_{FPt}p_{t}} - r_{f}}{\sqrt{\sum_{t}{\sum_{v}\lbrack {\min ( {{R_{{Pt}_{v}} - r_{f}},0} )} \rbrack^{2}}}p_{t}p_{v}}}$

where r_(f) is the forward-looking risk-free rate (the current one). Thedenominator is the downside target standard deviation based on thefactor model returns. It is also scaled by √{square root over (n_(Y))}.Note that for T=60 observations and m=51, the total number of returnrealizations is 3,060, enough points to expect an accuraterepresentation of downside frisk.

Forward-Looking VaR and CVaR for Portfolio Analysis

Using the return realizations obtained from the factor modelrepresentation with corresponding probabilities,

R _(Ptv) =R _(FPt)÷σ_(P) z _(v) p _(tv) =p _(t) p _(v)

we may compute the Value-at-Risk (VaR_(α)) by sorting the outcomes ofR_(Ptv) from the smallest to the largest value (maintaining thecorresponding p_(tv)). Call the sorted outcomes r_(j), j=1, . . . , Tmsuch that after sorting, (r₁, p₁) is the smallest outcome withcorresponding probability. (r₂, p₂) is the second smallest outcome withcorresponding probability, etc. One may then construct

P₁ P₂ P₃ P₄ . . . p₁ p₂ p₃ p₄ . . . r₁ r₂ r₃ r₄ . . .where P_(j) are the cumulative probabilities, i.e.,

$P_{j} = {\sum\limits_{k = 1}^{j}p_{k}}$

Now find the smallest index j* for which P_(j) equals or exceeds α.

If P_(j)*=a, then VaR_(α)=r_(j)*

If P_(j)*>α. then VaR_(α)=r_(j*÷1).

Then, the conditional Value-at-Risk may be calculated as

${CVaR}_{\alpha} = {{\frac{1}{\alpha}{\sum\limits_{j = 1}^{j_{\alpha}}{r_{j}p_{j}\mspace{14mu} {where}\mspace{14mu} j_{\alpha}}}} = { j \middle| {r(j)}  = {{VaR}_{\alpha}.}}}$

With this calculation, one obtains a forward-looking Value-at-Risk and aforward-looking conditional Value-at-Risk that are not based on assumingnormally distributed returns and, thus, reflect downside risk moreaccurately.

While the foregoing description has been with reference to particularexamples of the present invention, it will be appreciated by thoseskilled in the art that changes to these examples may be made withoutdeparting from the principles and spirit of the invention. Accordingly,the scope of the present invention can only be ascertained withreference to the appended claims.

What is claimed is:
 1. A method using a computer having a processorconfigured to execute instructions which when executed cause thecomputer to perform steps to manage a portfolio of financial assets toprovide large-scale portfolio optimization, including mean-varianceoptimization, expected utility maximization, and general mean-riskportfolio optimization, where asset returns are represented by a factormodel, comprising the steps of: selecting from multiple financial assetsa mix of a plurality of available financial assets comprising theportfolio of financial assets which is to be managed; selecting a factormodel which represents a distribution of asset returns for the pluralityof financial assets for a selected subsequent period of time for whichthe portfolio is to be managed, wherein asset returns in each period oftime t≥1 follow a factor model,{tilde over (R)} _(t) ={tilde over (F)} _(t) ^(T) {tilde over (V)}_(t)+{tilde over (ε)}_(t), where {tilde over (F)}_(t) is a k×n randommatrix of factor loadings, {tilde over (V)}_(t) is a random k-vector ofthe values of the factors including a mean vector, if the random valueof the first factor is defined as always having the value 1, and {tildeover (ε)}_(t) is a random n-vector of idiosyncratic returns where theidiosyncratic returns {tilde over (ε)}_(t) are multi-variate normallydistributed, {tilde over (ε)}_(t)=N(0, Σ_(t)), where the covarianceΣ_(t)=diag(σ_(it) ²), and {tilde over (ε)}_(t) is assumed independentlydistributed, between its components, respectively, and independentlydistributed with respect to {tilde over (V)}_(t); defining a firststatistical model applicable to macro-economic factor models by letting{tilde over (F)}_(t)=F be constant, {tilde over (V)}_(t) for t≥1 and{tilde over (ε)}_(t) for t≥0.1 each be independently and identicallydistributed random variables such that {tilde over (R)}_(t)=F^(T){tildeover (V)}_(t)+{tilde over (ε)}_(t) for t≥1 is an independently andidentically distributed random variable, so that observing at eachperiod t=1, . . . T an outcome R_(t), V_(t), and ε_(t) of {tilde over(R)}_(t), {tilde over (V)}_(t), and {tilde over (ε)}_(t), respectively,at period T+1, the current period at which a portfolio decision is tobe, made, the random vector of asset returns is{tilde over (R)} _(T+1) |{tilde over (R)} ₁ , . . . ,{tilde over (R)}_(T) =F ^(T) {tilde over (V)} _(T+1) |{tilde over (V)} ₁ , . . . ,{tildeover (V)} _(T)+{tilde over (ε)}_(T+1)|{tilde over (ε)}₁, . . . ,{tildeover (ε)}_(T). and based on independence,{tilde over (R)} _(T+1) =F ^(T) {tilde over (V)} _(T+1)+{tilde over(ε)}_(T+1) which results in{tilde over (R)}=F ^(T{tilde over (V)}+{tilde over (ε)}) by setting{tilde over (R)}≡{tilde over (R)}_(T+1), {tilde over (V)}≡{tilde over(V)}_(T+1) and {tilde over (ε)}≡{tilde over (ε)}_(T+1), therebysuppressing the time index for period T+1 such that {tilde over(ε)}=N(0, Σ), where Σ=diag(σ_(i) ²); defining a second statistical modelapplicable to fundamental factor models by letting {tilde over (F)}_(t)for t≥1 be a sequence of independently and identically distributedrandom variables and, conditional on {tilde over (F)}_(t), letting{tilde over (V)}_(t) and {tilde over (ε)}_(t) for t≥1 each beindependently and identically distributed random variables such that{tilde over (R)}_(t)|{tilde over (F)}_(t)={tilde over (F)}_(t)^(T){tilde over (V)}_(t)|{tilde over (F)}_(t)+{tilde over(ε)}_(t)|{tilde over (F)}_(t) is an independently and identicallydistributed random variable so that observing at each period t=1, . . .T an outcome R_(t), V_(t), F_(t), and ε_(t) of {tilde over (R)}_(t),{tilde over (V)}_(t), {tilde over (F)}_(t), and {tilde over (ε)}_(t),respectively, and an outcome F_(T+1) of {tilde over (F)}_(T+1), at thecurrent period T+1 at which a portfolio decision is to be made, therandom vector of asset returns is{tilde over (R)} _(T+1) |{tilde over (R)} ₁ ,{tilde over (F)} ₁ , . . .,{tilde over (R)} _(T) ,{tilde over (F)} _(T) ,{tilde over (F)} _(T+1)={tilde over (F)} _(T÷1) ^(T) {tilde over (V)} _(T+1) |{tilde over (V)}₁ ,{tilde over (F)} ₁ , . . . ,{tilde over (V)} _(T) ,{tilde over (F)}_(T) ,{tilde over (F)} _(T+1)+{tilde over (ε)}_(T+1)|{tilde over (ε)}₁,{tilde over (F)} ₁, . . . ,{tilde over (ε)}_(T) ,{tilde over (F)} _(T),{tilde over (F)} _(T+1) and based on independence,{tilde over (R)} _(T÷1) |{tilde over (F)} _(T+1) ={tilde over (F)}_(T+1) ^(T) {tilde over (V)} _(T+1) |{tilde over (F)} _(T+1)+{tilde over(ε)}_(T+1) |{tilde over (F)} _(T+1) and since at period T+1, an outcomeF_(T+1) of {tilde over (F)}_(T+1) is obtained{tilde over (R)}=F ^(T) {tilde over (V)}+{tilde over (ε)} by setting{tilde over (R)}≡{tilde over (R)}_(T+1)|{tilde over (F)}_(T+1)=F_(T+1),{tilde over (V)}≡{tilde over (V)}_(T+1)|{tilde over (F)}_(T+1)=F_(T+1),{tilde over (ε)}≡{tilde over (ε)}_(T+1)|{tilde over (F)}_(T+1)=F_(T+1),and F=F_(T+1), thereby suppressing the time index T+1 and the dependencyon the observed value F_(T÷1), of {tilde over (F)}_(T+1) such that{tilde over (ε)}=N(0, Σ), where E=diag(σ_(i) ²); maximizing expectedvalue of a function of the portfolio return defined as max EG({tildeover (R)}^(T)x) such that for mean-variance portfolio optimization theresult is max${{E\; {\overset{\sim}{R}}^{T}x} - {\frac{\gamma}{2}{{Risk}( {{\overset{\sim}{R}}^{T}x} )}}};$for utility maximization the result is max Eu(1+{tilde over (R)}^(T)x),and for mean-risk optimization the result is max${{E\; {\overset{\sim}{R}}^{T}x} - {\frac{\gamma}{2}{{var}( {{\overset{\sim}{R}}^{T}x} )}}},$approximating a unit normal random variable z by a discrete randomvariableζ=(z _(v) ,p _(v)) with realizations z_(v) occurring with probabilityp_(v), for v=1, . . . , m such that the continuous unit normaldistribution is represented by a histogram with properties that its meanis zero, E(ζ)=0, its variance is approximately one, E(ζ²)≈1, and itshigher moments match closely those of the unit normal distribution sothat for a sufficiently large number of discrete outcomes, the discreterepresentation closely approximates the unit normal distribution${{\lim\limits_{marrow\infty}\mspace{11mu} {\max\limits_{z}{{{(z)} - {(z)}}}}} = 0},$and the cumulative distribution function of the discrete approximation

(z) substantially corresponds to the cumulative distribution functionQ(z) of the unit normal distribution; utilizing a discrete approximationζ of the unit normal random variable z to determine the asset returns ofthe portfolio generated by the factor model for any realization z_(v) ofζ as{tilde over (R)} _(v)=(F ^(T) {tilde over (V)})^(T) x+σ(x)z _(v),defining the expected value as a function of the portfolio return as$\; {{{{EG}( {{\overset{\sim}{R}}_{v}(x)} )} = {\int_{- \infty}^{+ \infty}{\sum\limits_{v}{{G( {{( {F^{T}\overset{\sim}{V}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{v}{{dP}( \overset{\sim}{V} )}}}}};}$obtaining a discrete representation of the factor model returns asR _(iv)(x)=(F ^(T) V _(t))^(T) x+σ(x)z _(v), with associatedprobabilities P_(tv)=p_(t)p_(v); defining another discrete random vector

(x)=(R_(tv)(x),p_(tv)) with outcomes R_(tv)(x) and associatedprobability p_(tv) and utilizing a sample-average approximation definedby empirically observed outcomes V_(t) with corresponding probability$p_{t} = \frac{1}{T}$ to determine a conditional expectation, givenζ=z_(v), as${ {{EG}( {(x)} )} \middle| \zeta  = {\sum\limits_{i}{{G( {{( {F^{T}V_{t}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}}}},$where $p_{t} = \frac{1}{T}$ such that for a sufficiently large number Tof observations V_(t), EG(

(x))|ζ approximates EG({tilde over (R)}_(v)(x))|ζ, as EG(

(x))|ζ→EG({tilde over (R)}_(v)(x))|ζ as T→∞; and determining anexpectation EG(

(x)) as a multiple sum:${{{EG}( {\overset{\sim}{R}(x)} )} = {\sum\limits_{t}{\sum\limits_{v}{{G( {{( {F^{T}V_{t}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}p_{v}}}}},$and EG(

(x)) approximates EG({tilde over (R)}(x)), as EC(

(x))→EG({tilde over (R)}(x)) as T→∞ and as ζ approximates z; whereby fora general factor model representation of asset returns, portfolioreturns are expressed as a function of x as a random variable with adiscrete distribution representing a semi-parametric approximation,since the idiosyncratic component of the asset returns is representedparametrically and the factor explained component is representednon-parametrically and any expectation of functions of portfolio returnsthat may occur in a portfolio optimization model can therefore becomputed by multiple sums (over t and v); thereby making portfoliooptimization tractable and to facilitate solution.
 2. The method ofclaim 1 wherein a discrete approximation of the unit normaldistribution, obtained using optimization, is based on 51 equally spacedpoints between −5 and +5 and substantially corresponds to the unitnormal distribution wherein its, first 8 moments are mean=0:000000,variance=1.000000, skewness=0.000000, kurtosis=3.000000, m₅=0.000000,m₆=15.000000, m₇=0.000000, and m₈=105.000000 and a tail area of${\frac{1}{\sqrt{2\pi}}{\int_{- x}^{- 5}{{xe}^{- \frac{x^{- 2}}{2}}{dz}}}} = {{2.8665e} - 07}$of probability mass is all that is not captured on either side of theunit normal distribution and, utilizing 6 standard deviations, the onesided error is 9.8659e−10.
 3. The method of claim 1, further comprisingthe steps of: partitioning the factor explained returns F^(T){tilde over(V)} into a demeaned part F^(T){tilde over (V)}₀ and its mean vectorμ=F^(T)E{tilde over (V)}, where {tilde over (V)}=μ+{tilde over (V)}₀such that the factor-explained returns are {tilde over(R)}_(F)=μ+F^(T){tilde over (V)}₀ and the factor model returns areexpressed as {tilde over (R)}=μ+F^(T){tilde over (V)}₀+{tilde over (ε)}and observed outcomes of {tilde over (V)}₀ are denoted as V_(0t) andobserved outcome of {tilde over (R)}_(F) are denoted as R_(Ft); anddetermining expected utility maximization with a factor modelrepresentation of asset returns, comprising: defining the expectedutility maximizationmax E u(1+(F ^(T) {tilde over (V)}+{tilde over (ε)})^(T) x)Ax=b,l≤x≤h utilizing the semi-parametric discrete factor modelrepresentation of asset returns as:max Σ_(t)Σ_(v) u(1+R _(Ft) ^(T) x+σ(x)z _(v))p _(t) p _(v)Ax=b,l≤x≤h, where σ(x)=√{square root over (x²Σx)} to provide a discreteformulation with Tm realizations representing accurately the factormodel of returns, where for each outcome t there are in outcomesrepresenting the unit normal distribution multiplied with the nonlinearterm σ(x); obtaining gradients with respect to the decision variablesx_(i) as${\frac{\partial}{\partial x_{i}}{\sum\limits_{t}{\sum\limits_{v}{{u( {1 + {R_{Ft}^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}p_{v}}}}} = {\sum\limits_{t}{\sum\limits_{v}{{u^{\prime}( {1 + {R_{Ft}^{T}x} + {{\sigma (x)}z_{v}}} )}( {{\overset{\sim}{R}}_{Fi} + {\frac{1}{\sigma (x)}\sigma_{i}^{2}x_{i}z_{v}}} )p_{i}p_{v}}}}$for each i=1 . . . , n, where σ(x)=√{square root over (x^(T)Σx)}; andutilizing a gradient-based nonlinear optimization program running on aprocessor to determine the expected utility maximization.
 4. The methodof claim 3, further comprising the steps of: obtaining equilibriumreturns (de such that the expected utility maximizationmax E u(1+(μ_(e) +F ^(T) V _(0t)+ε)^(T) x)e ^(T) x=1 for the utility function of a benchmark u=u_(B) results in abenchmark portfolio x_(B):${\mu_{ei} = {{\mu_{B} - {\frac{E\; {u_{B}^{\prime}( {1 + \mu_{B} + \eta_{Bt} + \epsilon_{B}} )}( {\eta_{ti} - \eta_{Bt} + \epsilon_{i} - \epsilon_{B}} )}{E\; {u_{B}^{\prime}( {1 + \mu_{B} + \eta_{Bt} + \epsilon_{B}} )}}i}} = 1}},\ldots \mspace{14mu},n,$where Θ_(Bt)=F^(T)V_(0t) ^(T)x₈ is the demeaned factor-explained return;and determining the expectations for the equilibrium returns utilizing asemi-parametric discrete representation as${E\; {\mu_{B}^{\prime}( {1 + \mu_{B} + \eta_{Bt} + {\sigma_{B}z_{1}}} )}( {\eta_{ti} - \eta_{Bt} + {\sigma_{i}z_{2}} - {\sigma_{B}z_{1}}} )} = {\sum\limits_{t}{\sum\limits_{v_{1}}{\sum\limits_{v_{2}}{{u_{B}^{\prime}( {1 + \mu_{B} + \eta_{Bt} + {\sigma_{B}z_{v_{1}}}} )}( {\eta_{ti} - \eta_{Bt} + {\sigma_{i}( {{c_{iB}z_{v_{1}}} + {\sqrt{1 - c_{iB}^{2}}z_{v_{2}}}} )} - {\sigma_{B}z_{v_{1}}}} )p_{t}p_{v_{1}}p_{v_{2}}}}}}$     and$\mspace{79mu} {{E\; {\mu_{B}^{\prime}( {1 + \mu_{B} + \eta_{Bt} + {\sigma_{B}z_{v}}} )}} = {\sum\limits_{t}{\sum\limits_{v}{{u_{B}^{\prime}( {1 + \mu_{B} + \eta_{Bt} + {\sigma_{B}z_{v}}} )}{\underset{\_}{p}}_{t}p_{v}}}}}$by Ling discrete approximations ζ₁=(z_(v) ₁ ,p_(v) ₁ ), ζ₂=(z_(v) ₂,p_(v) ₂ ), and ζ₃=(z_(v),p_(v)) of the independent unit normal randomvariables z₁, z₀, and z, respectively.
 5. The method of claim 4, furthercomprising the step of: utilizing μ_(e) to determineμ_(e)=μ−μ_(e), where μ=F^(T)E{tilde over (V)} is the mean value of thefactor explained return to determine the expected utility maximizationportfolio model based on semi-parametric discrete factor modelrepresentation of asset returns as$\max {\sum\limits_{t}{\sum\limits_{v}{{u( {1 + {( {{\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e} + {F^{T}V_{0\; t}}} )^{T}x} + {{\sigma (x)}z_{v}}} )}p_{t}p_{v}}}}$Ax = b, l ≤ x ≤ h, where σ(x)=√{square root over (x^(T)Σx)} and γ_(e)scales the conditional expected returns.
 6. The method of claim 1,further comprising determining mean-variance portfolio optimization witha factor model representation of asset returns, comprising the steps of:utilizing a factor-model-based covariance representation${\max \; {E( {F^{T}\overset{\sim}{V}} )}^{T}x} - {\frac{\gamma}{2}{x^{T}( {{F^{T}M_{\overset{\sim}{V}}F} + \sum} )}x}$Ax = b, l ≤ x ≤ h where M_({tilde over (V)}) is a=k×k covariance matrixof the factors and Σ=diag(σ_(i) ²) is the diagonal matrix ofidiosyncratic variance; and determining the variance σ_(i) ² of the i-thindependent error term {tilde over (ε)}_(i) using the semi-parametricdiscrete factor model representation of asset, returns asmax μ^(T) x−γ/2Σ_(t)Σ_(v)((F ^(T) V _(0t))^(T) x+σ(x)u _(v))² p _(t) p_(v)Ax=b,l≤x≤h, where σ(x)=√{square root over (x^(T)Σx)} to represent ascenario formulation, Of mean-variance having Tm scenarios representinga covariance structure using a gradient-based nonlinear program runningon a processor.
 7. The method of claim 6, further comprising the stepsof: obtaining equilibrium returns μ_(e) utilizing the factor model asμ_(e)=γ_(B)(F ^(T) M _({tilde over (V)}) F+Σ)x _(B)to determineμ_(c)=μ−μ_(e); and determining the mean-variance portfolio optimizationas${{\max ( {{\frac{1}{\gamma_{c}}\mu_{c}} + \mu_{e}} )}^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{( {{( {F^{T}V_{0\; t}} )^{T}x} + {{\sigma (x)}z_{v}}} )^{2}p_{t}p_{v}}}}}$Ax = b, l ≤ x ≤ h
 8. The method of claim 1, further comprisingdetermining mean-risk portfolio optimization with a factor modelrepresentation of asset returns, comprising the steps of: defining theprobability of the asset returns of a portfolio x asΨ({tilde over (R)} ^(T) x,W)=∫_({tilde over (R)}) _(T) _(x≤−W) p({tildeover (R)})d{tilde over (R)}, where {tilde over (R)}^(T)x does not exceeda threshold W; defining Value-at-Risk VaR_(α)({tilde over (R)}^(T)x) forcontinuous distribution functions asVaR_(α)({tilde over (R)} ^(T) x)=min{W: Ψ({tilde over (R)} ^(T)x,W)≤α)}; defining Conditional-Value-at-Risk CVaR_(α)({tilde over(R)}^(T)x) for continuous distribution functions as${{{CVaR}_{\alpha}( {{\overset{\sim}{R}}^{T}x} )} = {\frac{1}{\alpha}{\int_{{{\overset{\sim}{R}}^{T}x} \leq {- {{VaR}_{\alpha}{({{\overset{\sim}{R}}^{T}x})}}}}^{\;}{{- {\overset{\sim}{R}}^{T}}{{xp}( \overset{\sim}{R} )}d\overset{\sim}{R}}}}};$utilizing the definitions for Value-at-Risk andConditional-Value-at-Risk and a function${F_{\alpha}( {{{\overset{\sim}{R}}^{T}x},W} )} = {W + {\frac{1}{\alpha}{E( {{{- {\overset{\sim}{R}}^{T}}x} - W} )}^{+}}}$to define the mean-risk portfolio optimization withConditional-Value-at-Risk as the risk measure as${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}{{CVaR}_{\alpha}( {{\overset{\sim}{R}}^{T}x} )}}$Ax = b, l ≤ x ≤ h; determining${{CVaR}_{\alpha}( {{\overset{\sim}{R}}^{T}x} )} = {\min\limits_{W,x}{F_{\alpha}( {{{\overset{\sim}{R}}^{T}x},W} )}}$using the semi-parametric and discrete representation of the factormodel returns as${\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}( {W + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}p_{t}p_{v}}}}}} )}$(F^(T)V_(t))^(T)x + σ(x)z_(v) + u_(tv) + W ≥ 0, u_(tv) ≥ 0, ∀t, vAx = b, l ≤ x ≤ h, where σ(x)=√{square root over (x^(T)Σx)}; andutilizing a gradient-based nonlinear optimization program running on aprocessor to determine${W^{*} + {\frac{1}{\alpha}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{*}p_{t}p_{v}}}}}} = {{CVaR}_{\alpha}( {R^{T}x^{*}} )}$as the optimal Conditional-Value-at-Risk value and W*=VaR_(α)({tildeover (R)}^(T)x*) as the Value-at-Risk value of the returns of theoptimal portfolio.
 9. The method of claim 1, further comprisingdetermining mean-risk portfolio optimization with a factor modelrepresentation of asset returns, comprising the steps of: definingmean-risk portfolio optimization with mean-absolute-deviation (MAM) as arisk measure as${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}{{MAD}( {{\overset{\sim}{R}}^{T}x} )}}$Ax = b, l ≤ x ≤ h, where${{{MAD}( {{\overset{\sim}{R}}^{T}x} )} = {E{{{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}}}}};$utilizing the semi-parametric and discrete representation of the factorasset returns as${{{\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{( {u_{tv}^{+} + u_{tv}^{-}} )p_{t}p_{v}}}}} - {( {F^{T}V_{0\; t}} )^{T}x} - {{\sigma (x)}z_{v}} + u_{tv}^{+} - u_{tv}^{-}} = 0},u_{tv}^{+},{u_{tv}^{-} \geq 0},{\forall t},v$Ax = b, l ≤ x ≤ h, where σ(x)=√{square root over (x^(T)Σx)}; andutilizing a gradient-based nonlinear optimization program running on aprocessor to determine mean-risk portfolio optimization with MAD as therisk measure.
 10. The method of claim 1, further comprising determiningmean-risk portfolio optimization with a factor model representation ofasset returns, comprising the steps of: defining mean-risk portfoliooptimization with mean-absolute-moment (MAM) as the risk measure as${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}{{MAM}_{q}( {{\overset{\sim}{R}}^{T}x} )}}$Ax = b, l ≤ x ≤ h, where${{{MAM}_{q}( {{\overset{\sim}{R}}^{T}x} )} = {E{{{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}}}^{q}}},{{q > 1};}$utilizing the semi-parametric and discrete representation of the factormodel returns as${{{\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{( {u_{tv}^{+} + u_{tv}^{-}} )p_{t}p_{v}}}}} - {( {F^{T}V_{0\; t}} )^{T}x} - {{\sigma (x)}z_{v}} + u_{tv}^{-} - u_{tv}^{-}} = 0},u_{tv}^{+},{u_{tv}^{-} \geq 0},{\forall t},v$Ax = b, l ≤ x ≤ h where σ(x)=√{square root over (x^(T)Σx)}; andutilizing a gradient-based nonlinear optimization program running on aprocessor to determine mean-risk portfolio optimization with IMAM as therisk measure.
 11. The method of claim 1, further comprising determiningmean-risk portfolio optimization with a factor model representation ofasset returns, comprising the steps of: defining mean-risk portfoliooptimization with semi-variance (σ_(semi) ²) as the risk measure as${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}{\sigma_{semi}^{2}( {{\overset{\sim}{R}}^{T}x} )}}$Ax = b, l ≤ x ≤ h, where${{\sigma_{semi}^{2}( {{\overset{\sim}{R}}^{T}x} )} = {E\; {\min ( {{{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}},0} )}^{2}}},$and where only portfolio return outcomes smaller than the expectedreturn are considered in the variance determination; utilizing thesemi-parametric and discrete representation of the factor asset returnsas${{{\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{2}p_{t}{p_{v}( {F^{T}V_{0\; t}} )}^{T}x}}}} + {{\sigma (x)}z_{v}} + u_{tv}} \geq 0},{u_{tv} \geq 0},{\forall t},v$Ax = b, l ≤ x ≤ h, where σ(x)=√{square root over (x^(T)Σx)}, end whereV_(0t) is the demeaned part of the factor return; and utilizing agradient-based nonlinear optimization program running on a processor todetermine mean-risk portfolio optimization with semi-variance (σ_(semi)²) as, the risk measure.
 12. The method of claim 1, further comprisingdetermining mean-risk portfolio optimization with a factor modelrepresentation of asset returns, comprising the steps of: definingmean-risk portfolio optimization with lower partial moment (LPM_(q)w) ofthe power q as the risk measure as${{\max ( {E\overset{\sim}{R}} )}^{T}x} - {\frac{\gamma}{2}{{Risk}( {{\overset{\sim}{R}}^{T}x} )}}$Ax = b, l ≤ x ≤ h, where${{{LPM}_{qW}( {{\overset{\sim}{R}}^{T}x} )} = {E( {- {\min ( {{{{\overset{\sim}{R}}^{T}x} - W},0} )}} )}^{q}},{q \geq 1},$where W is a predefined value of return and risk is considered as theexpected value of the negative under-performance with respect to a fixedlevel W raised to the power of q; utilizing the semi-parametric anddiscrete representation of the factor asset returns as${{{\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{\sum\limits_{t}{\sum\limits_{v}{u_{tv}^{q}p_{t}{p_{v}( {F^{T}V_{t}} )}^{T}x}}}} + {{\sigma (x)}z_{v}} + u_{tv}} \geq W},{u_{tv} \geq 0},{\forall t},v$Ax = b, l ≤ x ≤ h where σ(x)=√{square root over (x^(T)Σx)} and where Wis a predefined constant; for q=1, utilizing a linear optimizationprogram running on a processor to determine mean-risk portfoliooptimization with the lower partial moment (LPM_(q)W) of the power q asthe risk measure; and for q>1, utilizing a gradient-based nonlinearoptimization program running on a processor to determine mean-riskportfolio optimization with the lower partial moment (LPM_(q)w) of thepower q as the risk measure.
 13. The method of claim 1, furthercomprising determining mean-variance portfolio optimization with afactor model representation of asset returns having a risk constraintwith CVaR as the risk measure, comprising the steps of: defining a CVaRconstraint as part of a mean-variance portfolio, optimization as${\max \mspace{14mu} \mu^{T}x} - {\frac{\gamma}{2}{x^{T}( {{F^{T}M_{\overset{\sim}{V}}F} + \sum} )}x}$Ax = b, l ≤ x ≤ h${{W + {\frac{1}{\alpha}{\sum_{t}\; {\sum_{v}\; {u_{tv}p_{t}p_{v}}}}}} \leq {{{\rho ( {F^{T}V_{t}} )}^{T}x} + {{\sigma (x)}z_{v}} + u_{tv} + W} \geq 0},{u_{tv} \geq 0},{\forall t},v,$where ρ is a given maximal Value-at-Risk level; and utilizing agradient-based nonlinear optimization program running on a processor todetermine mean-risk portfolio optimization having a risk constraint withCVaR as the risk measure.
 14. The method of claim 1, further comprisingdetermining expected utility optimization with a factor modelrepresentation of asset returns having a risk constraint withConditional-Value-at-Risk (CVaR) as the risk measure, comprising thesteps of: defining a CVaR constraint as part of the expected utilitymaximization asmax ∑_(t) ∑_(v) u(1 + R_(Ft)^(T)x + σ(x)z_(v))p_(t)p_(v)Ax = b, l ≤ x ≤ h${{W + {\frac{1}{\alpha}{\sum_{t}\; {\sum_{v}\; {u_{tv}p_{t}p_{v}}}}}} \leq {{{\rho ( {F^{T}V_{t}} )}^{T}x} + {{\sigma (x)}z_{v}} + u_{tv} + W} \geq 0},{u_{tv} \geq 0},{\forall t},v,$where ρ is a given maximal Value-at-Risk level; and utilizing agradient-based nonlinear optimization program running on a processor todetermine expected utility maximization having a risk constraint withCVaR as the risk measure.
 15. The method of claim 1, further comprisingthe steps of: defining skewness as${{Skew} = \frac{{E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{3}}{( {E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{2} )^{\frac{3}{2}}}};$constraining skewness to be greater than or equal to a given lower boundSkew_(l); defining a variable u_(tv) representing the demeaned portfolioreturns asv _(tv)=(F ^(T) V _(0t))^(T) x+σ(x)z _(v), where σ(x)=√{square root over(x^(T)Σx)}; and utilizing the factor model asset returns and itsdiscrete representation as${{\sum\limits_{t}\; {\sum\limits_{v}\; {u_{tv}^{3}p_{t}p_{v}}}} - {{Skew}_{l}( {\sum\limits_{t}\; {\sum\limits_{v}\; {u_{tv}^{2}p_{t}p_{v}}}} )}^{\frac{3}{2}}} \geq 0$to determine a skewness constraint, wherein the skewness constraintsimplifies if skewness is constrained to be nonnegative (Skew≥0) as${{\sum\limits_{t}\; {\sum\limits_{v}\; {u_{tv}^{3}p_{t}p_{v}}}} \geq 0};{and}$applying the skewness constraint to the portfolio asset returns {tildeover (R)}^(Tx).
 16. The method of claim 1, further applying a kurtosisconstraint on the distribution of portfolio returns {tilde over(R)}^(T)x, comprising the steps of: defining kurtosis as${{Kurt} = \frac{{E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{4}}{( {E( {{{\overset{\sim}{R}}^{T}x} - {E( {{\overset{\sim}{R}}^{T}x} )}} )}^{2} )^{2}}};$constraining kurtosis to be less than or equal to a given upper boundKurt_(h); defining a variable v_(tv) representing the demeaned portfolioreturns asv _(tv)=(F ^(T) V _(0t))^(T) x+σ(x)z _(v), where J(x)=√{square root over(x^(T)Σx)}; and utilizing the factor model asset returns and itsdiscrete representation as${{\sum\limits_{t}\; {\sum\limits_{v}\; {u_{tv}^{4}p_{t}p_{v}}}} - {{Kurt}_{h}( {\sum\limits_{t}\; {\sum\limits_{v}\; {u_{tv}^{2}p_{t}p_{v}}}} )}^{2}} \leq 0$to determine a kurtosis constraint; and applying the kurtosis constraintto the portfolio asset returns {tilde over (R)}^(T)x.
 17. The method ofclaim 1, further comprising, the steps of: incorporating at least onederivative security

as part of the portfolio of financial assets, wherein the at least onederivative security is selected from the group of derivative securitiesconsisting of options, forwards, futures, and swaps, whose price dependson the price of an underlying security, with an asset return redepending on the underlying portfolio xu represented asr _(l) =f _(l)({tilde over (R)} ^(T) x _(U)), where f_(l)(·) is thereturn generating function of the at least one derivative security

and {tilde over (R)}_(U)={tilde over (R)}^(T)x_(U); defining f as then_(d)-vector of return generating functions of derivative securities andy as the n_(d)-vector of holdings of the derivative securities;maximizing the expectation of a function G of asset returns of theportfolio comprising the derivative securities asmax EG({tilde over (R)} ^(T) x+f ^(T)({tilde over (R)} ^(T) x _(U))y)subject to portfolio, constraints on n n_(d) assets; utilizing thesemi-parametric discrete factor model representation of asset returns torepresent the expectation as${\max {\sum\limits_{t}\; {\sum\limits_{v_{1}}\; {\sum\limits_{v_{2}}\; {{G( {{R_{Ft}^{T}x} + {{\sigma (x)}z_{v_{1}}} + {{f^{T}( {{R_{Ft}^{T}x_{U}} + {\sigma_{U}( {{c_{xU}z_{v_{1}}} + {\sqrt{1 - c_{xU}^{2}}z_{v_{2}}}} )}} )}y}} )}p_{t}p_{v\; 1}p_{v\; 2}}}}}},$where ζ₁=(z_(v) ₁ , p_(v) ₁ ) and ζ₂=(z_(v) ₂ ,p_(v) ₂ ) are discreteapproximations of the independent unit normal random variables z₁ andz₂; and optimizing the expectation subject to portfolio constraints onall n+n_(d) assets.
 18. The method of claim 1, further comprising thesteps of: defining a forward-looking Sortino Ratio of the portfoliox_(P) as${{SoR} = {\sqrt{n_{Y}}\frac{E( {{{\overset{\sim}{R}}^{T}x_{P}} - r_{f}} )}{\sqrt{{E\lbrack {\min ( {{{{\overset{\sim}{R}}^{T}x_{P}} - r_{f}},0} )} \rbrack}^{2}}}}};$utilizing the semi-parametric and discrete representation of the factormodel asset returns for the portfolio x_(P),R _(Ptv) =R _(Ft) ^(T) x _(P)+σ(x _(P))z _(v); and determining theforward-looking Sortino Ratio of the portfolio x_(P) as${{SoR} = {\sqrt{n_{Y}}\frac{{\sum_{t}\; {R_{FPt}p_{t}}} - r_{f}}{\sqrt{\sum_{t}\; {\sum_{v}\; {\lbrack {\min ( {{{R_{Ptv}z_{v}} - r_{f}},0} )} \rbrack^{2}p_{t}p_{v}}}}}}},$where r_(j) is the forward-looking risk-free rate and n_(Y) is thenumber of observations per year used for the estimation of the factormodel; wherein Tm forward-looking return realizations with correspondingprobability p_(t)p_(v), are used to represent the forward-looking assetreturns distribution of the portfolio x_(P); and wherein the SortinoRatio is the scaled ratio of the excess returns over the risk-free rater_(f) divided by the downside target standard deviation and is expressedin annual terms.
 19. The method of claim 1, further comprising the stepsof: defining a downside target standard deviation corresponding to thesquare root of the lower partial moment of order 2 of {tilde over(R)}^(T)x_(P) for portfolio asset returns {tilde over (R)}^(T)x_(P) as√{square root over (E[min({tilde over (R)} ^(T) x _(P) −r _(f),0)]²)};utilizing the semi-parametric and discrete representation of the factormodel asset returns for the portfolio x_(P)R _(Ptv) =R _(Ft) ^(T) x _(P)+σ(x _(P))z _(v); and determining thedownside target standard deviation as${\sqrt{\sum\limits_{t}\; {\sum\limits_{v}\; \lbrack {\min ( {{R_{Ptv} - r_{f}},0} )} \rbrack^{2}}}p_{t}p_{v}},$where r_(f) is the forward-looking risk-free rate, and the downsidetarget standard deviation may be annualized by multiplying by √{squareroot over (n_(Y))}.
 20. The method of claim 1, further comprising thesteps of: utilizing the semi-parametric and discrete representation offactor model asset returns with its corresponding probabilities,R _(Ptv) =R _(FPt)+σ_(P) z _(v) p _(tv) =p _(t) p _(v); determining aValue-at-Risk (VaR_(α)) by sorting the outcomes of R_(Ptv), from thesmallest to the largest value, maintaining the corresponding p_(tv); andutilizing the sorted outcomes r_(j), j=1, . . . , Tm, where j=1 is thesmallest value; and determining the cumulative probabilities P_(j) as${P_{j} = {\sum\limits_{k = 1}^{j}\; p_{k}}},$ where the smallestindex j* for which P_(j) equals or exceeds α is found: if P_(j*)=α, thenVaR_(α)=r_(j*). and if P_(j*)>α, then VaR_(α)=r_(j*+1).
 21. The methodof claim 1, further comprising the step of: determining theConditional-Value-at-Risk as${{CVaR}_{\alpha} = {\frac{1}{\alpha}{\sum\limits_{j = 1}^{j_{\alpha}}\; {r_{j}p_{j}}}}},$where j_(α)=j|r(j)=VaR_(α). by summing the sorted returns up to theindex j for which the return r(j) is the VaR_(α) value and by dividingthe sum by α.
 22. The method of claim 1 wherein the factor model ofasset returns is defined for asset risk premia {tilde over(R)}_(t)−r_(ft)e, corresponding to excess returns over the risk-freerate, r_(ft), such that at each rime t≥1. risk premia follow the factormodel:({tilde over (R)} _(t) −r _(ft) e)={tilde over (F)} _(t) ^(T) {tildeover (V)} _(t)+{tilde over (ε)}_(t), where r_(ft) is the risk-free rateat period t, and e is an n-vector of ones.